Quantum-state engineering with Josephson-junction devices

Transcription

1 Quantum-state engineering with Josephson-junction devices Yuriy Makhlin Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-7618 Karlsruhe, Germany and Landau Institute for Theoretical Physics, Kosygin st., Moscow, Russia Gerd Schön Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-7618 Karlsruhe, Germany and Forschungszentrum Karlsruhe, Institut für Nanotechnologie, D-7601 Karlsruhe, Germany Alexander Shnirman Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-7618 Karlsruhe, Germany (Published 8 May 001) REVIEWS OF MODERN PHYSICS, VOLUME 73, APRIL 001 Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations. CONTENTS I. Introduction 357 II. Josephson Charge Qubit 359 A. Superconducting charge box as a quantum bit 359 B. Charge qubit with tunable coupling 361 C. Controlled interqubit coupling 36 D. Experiments with Josephson charge qubits 364 E. Adiabatic charge manipulations 365 III. Qubits Based on the Flux Degree of Freedom 366 A. Josephson flux (persistent current) qubits 367 B. Coupling of flux qubits 369 C. Quiet superconducting phase qubits 369 D. Switches 371 IV. Environment and Dissipation 371 A. Identifying the problem 371 B. Spin-boson model 37 C. Several fluctuating fields and many qubits 374 D. Dephasing in charge qubits 374 E. Dephasing in flux qubits 376 V. The Quantum Measurement Process 377 A. General concept of quantum measurements 377 B. Single-electron transistor as a quantum electrometer 378 C. Density matrix and description of measurement 380 D. Master equation 381 E. Hamiltonian-dominated regime 38 F. Detector-dominated regime 385 G. Flux measurements 386 H. Efficiency of the measuring device 386 I. Statistics of the current and the noise spectrum 388 J. Conditional master equation 389 VI. Conclusions 390 Acknowledgments 39 Appendix A: An Ideal Model The model Hamiltonian 39. Preparation of the initial state Single-qubit operations Two-qubit operations 393 Appendix B: Quantum Logic Gates and Quantum Algorithms Single- and two-qubit gates 393. Quantum Fourier transformation Quantum computation and optimization 394 Appendix C: Charging Energy of a Qubit Coupled to a Set 395 Appendix D: Derivation of the Master Equation 395 References 397 I. INTRODUCTION The interest in macroscopic quantum effects in lowcapacitance Josephson-junction circuits has persisted for /001/73()/357(44)/$ The American Physical Society

2 358 Makhlin, Schön, and Shnirman: Quantum-state engineering many years. One of the motivations was to test whether the laws of quantum mechanics applied in macroscopic systems, in a Hilbert space spanned by macroscopically distinct states (Leggett, 1987). The degrees of freedom studied were the phase difference of the superconducting order parameter across a junction or the flux in a superconducting quantum interference device (SQUID) ring geometry. Various quantum phenomena, such as macroscopic quantum tunneling and resonance tunneling, were demonstrated (see, for example, Voss and Webb, 1981; Martinis et al., 1987; Rouse et al., 1995). On the other hand, despite experimental efforts (e.g., Tesche, 1990), coherent oscillations of the flux between two macroscopically distinct states (macroscopic quantum coherence) had not been observed. The field received new attention recently, after it was recognized that suitable Josephson devices might serve as quantum bits (qubits) in quantum information devices and that quantum logic operations 1 could be performed by controlling gate voltages or magnetic fields (see, for example, Bouchiat, 1997; Shnirman et al., 1997; Averin, 1998; Ioffe et al., 1999; Makhlin et al., 1999; Mooij et al., 1999; Nakamura et al., 1999). In this context, as well as for other conceivable applications of quantum-state engineering, the experimental milestones are the observation of quantum superpositions of macroscopically distinct states, of coherent oscillations, and of entangled quantum states of several qubits. For Josephson devices the first successful experiments have been performed. These systems can be fabricated by established lithographic methods, and the control and measurement techniques are quite advanced. They further exploit the coherence of the superconducting state, which helps to achieve sufficiently long phase coherence times. Two alternative realizations of quantum bits have been proposed, based on either charge or phase (flux) degrees of freedom. In the former, the charge in lowcapacitance Josephson junctions is used as a quantum degree of freedom, with basis states differing by the number of Cooper-pair charges on an island. These devices combine the coherence of Cooper-pair tunneling with the control mechanisms developed for singlecharge systems and Coulomb-blockade phenomena. The manipulations can be accomplished by switching gate voltages (Shnirman et al., 1997); designs with controlled interqubit couplings were proposed (Averin, 1998; Makhlin et al., 1999). Experimentally, the coherent tunneling of Cooper pairs and the related properties of quantum-mechanical superpositions of charge states have been demonstrated (Bouchaiat, 1997; Nakamura et al., 1997). Most spectacular are recent experiments of Nakamura et al. (1999) in which the quantum-coherent oscillations of a Josephson charge qubit prepared in a 1 Since computational applications are widely discussed, we frequently employ here and below the terminology of quantum information theory, referring to a two-state quantum system as a qubit and denoting unitary manipulations of its quantum state as quantum logic operations or gates. superposition of eigenstates were observed in the time domain. We describe these systems, concepts, and results in Sec. II. The alternative realization is based on the phase of a Josephson junction or the flux in a ring geometry near a degeneracy point as a quantum degree of freedom (see, for example, Ioffe et al., 1999; Mooij et al., 1999). In addition to the earlier experiments, in which macroscopic quantum tunneling had been observed (Voss and Webb, 1981; Martinis et al., 1987; Rouse et al., 1995), the groups in Delft and Stony Brook (Friedman et al., 000; van der Wal et al., 000) recently demonstrated by spectroscopic measurements the flux qubit s eigenenergies; they observed eigenstates that are superpositions of different flux states, and new efforts are being made to observe the coherent oscillation of the flux between degenerate states (Cosmelli et al., 1998; Mooij et al., 1999; Friedman et al., 000). We shall discuss the quantum properties of flux qubits in Sec. III. To make use of the quantum coherent time evolution it is crucial to find systems with intrinsically long phase coherence times and to minimize external sources of dephasing. The latter can never be avoided completely since, in order to perform the necessary manipulations, one has to couple to the qubits, for instance, by attaching external leads. Along the same channels as the signal (e.g., gate voltages) noise also enters the system. However, by operating at low temperatures and choosing suitable coupling parameters, one can keep these dephasing effects at an acceptable level. We provide estimates of the phase coherence time in Sec. IV. In addition to controlled manipulations of qubits, quantum measurement processes are needed, for example, to read out the final state of the system. In our quantum mechanics courses we learned to express the measurement process as a wave-function collapse, i.e., as a nonunitary projection, which reduces the quantum state of the qubit to one of the possible eigenstates of the observed quantity with state-dependent probabilities. However, in reality any measurement is performed by a device that itself is realized by a physical system, suitably coupled to the measured quantum system and with a macroscopic readout variable. Its presence, in general, disturbs the quantum manipulations. Therefore the dissipative processes that accompany the measurement should be switched on only when needed. An example is provided by a normal-state singleelectron transistor (SET) coupled capacitively to a single-cooper-pair box. This system is widely used as an electrometer in classical single-charge systems. We describe in Sec. V how a SET can also be used to read out the quantum state of a charge qubit. For this purpose we study the time evolution of the coupled system s density matrix (Shnirman and Schön, 1998). During quantum manipulations of the qubit the transport voltage of the SET is turned off, in which case it acts only as an extra capacitor. To perform the measurement the transport voltage is turned on. In this stage the dissipative current through the transistor rapidly dephases the state of the qubit. This current also provides the macroscopic read-

3 Makhlin, Schön, and Shnirman: Quantum-state engineering 359 out signal for the quantum state of the qubit. However, it requires a longer measurement time until the noisy signal resolves different qubit states. Finally, on the still longer mixing time scale, the measurement process itself destroys the information about the initial quantum state. Many results and observations made in the context of the normal-state single-electron transistor also apply to other physical systems, e.g., a superconducting SET (SSET) coupled to a charge qubit (Averin, 000b; Cottet et al., 000) or a dc SQUID monitoring as a quantum magnetometer the state of a flux qubit (see, for example, Mooij et al., 1999; Averin, 000b; Friedman et al., 000). The results can also be compared to the nonequilibrium dephasing processes discussed theoretically (Aleiner et al., 1997; Gurvitz, 1997; Levinson, 1997) and demonstrated experimentally by Buks et al. (1998). One of the motivations for quantum-state engineering with Josephson devices is their potential application as logic devices and quantum computing. By exploiting the massive parallelism of the coherent evolution of superpositions of states, quantum computers could perform certain tasks that no classical computer could do in acceptable times (Bennett, 1995; DiVincenzo, 1995; Barenco, 1996; Aharonov, 1998). In contrast to the development of physical realizations of qubits and gates, i.e., the hardware, the theoretical concepts of quantum computing, the software, are already rather advanced. As an introduction, and in order to clearly define the goals, we present in Appendix A an ideal model Hamiltonian with sufficient control to perform all the needed manipulations. (We note that the Josephsonjunction devices come rather close to this ideal model.) In Appendix B we show by a few representative examples how these manipulations can be combined for useful computations. Various other physical systems have been suggested as possible realizations of qubits and gates. They are discussed in much detail in a recent Fortschritte der Physik special issue entitled Experimental Proposals for Quantum Computation (Braunstein and Lo, 000). In some systems quantum manipulations of a few qubits have already been demonstrated experimentally. These include ions in electromagnetic traps manipulated by laser irradiation (Cirac and Zoller, 1995; Monroe et al., 1995), nuclear magnetic resonance (NMR) on ensembles of molecules in liquids (Cory et al., 1997; Gershenfeld and Chuang, 1997) and cavity QED systems (Turchette et al., 1995). In comparison, solid-state devices, including the mentioned Josephson systems, are more easily embedded in electronic circuits and scaled up to large registers. Ultrasmall quantum dots with discrete levels and, in particular, spin degrees of freedom embedded in nanostructured materials are candidates as well. They can be manipulated by tuning potentials and barriers (Kane, 1998; Loss and DiVincenzo, 1998). Because of the difficulties of controlled fabrication, their experimental realization is still at a very early stage. FIG. 1. A Josephson charge qubit in its simplest design formed by a superconducting single-charge box. II. JOSEPHSON CHARGE QUBIT A. Superconducting charge box as a quantum bit In this section we describe the properties of lowcapacitance Josephson junctions, in which the charging energy dominates over the Josephson coupling energy, and discuss how they can be manipulated in a quantumcoherent fashion. Under suitable conditions they provide physical realizations of qubits with two states differing by one Cooper pair charge on a small island. The necessary one-bit and two-bit gates can be performed by controlling applied gate voltages and magnetic fields. Different designs will be presented that differ not only in complexity, but also in the accuracy and flexibility of the manipulations. The simplest Josephson-junction qubit is shown in Fig. 1. It consists of a small superconducting island ( box ) with n excess Cooper-pair charges (relative to some neutral reference state), connected by a tunnel junction with capacitance C J and Josephson coupling energy E J to a superconducting electrode. A control gate voltage V g is coupled to the system via a gate capacitor C g. Suitable values of the junction capacitance, which can be fabricated routinely by present-day technologies, are in the range of femtofarad and below, C J F, while the gate capacitances can be chosen still smaller. The relevant energy scale, the single-electron charging energy E C e /(C g C J ), which depends on the total capacitance of the island, is then in the range of a Kelvin or above, E C 1 K. The Josephson coupling energy E J is proportional to the critical current of the Josephson junction (see, for example, Tinkham, 1996). Typical values considered here are in the range of 100 mk. We choose a material such that the superconducting energy gap is the largest energy in the problem, larger even than the single-electron charging energy. In this Throughout this review we frequently use temperature units for energies.

4 360 Makhlin, Schön, and Shnirman: Quantum-state engineering case quasiparticle tunneling is suppressed at low temperatures, and a situation can be reached in which no quasiparticle excitation is found on the island. 3 Under these conditions only Cooper pairs tunnel coherently in the superconducting junction, and the system is described by the Hamiltonian H4E C nn g E J cos. (.1) Here n is the number operator of (excess) Cooper-pair charges on the island, and, the phase of the superconducting order parameter of the island, is its quantummechanical conjugate, ni /(). The dimensionless gate charge, n g C g V g /e, accounts for the effect of the gate voltage and acts as a control parameter. Here we consider systems in which the charging energy is much larger than the Josephson coupling energy, E C E J. In this regime a convenient basis is formed by the charge states, parametrized by the number of Cooper pairs n on the island. In this basis the Hamiltonian (.1) reads H n 4E C nn g nn 1 E Jnn1n1n. (.) For most values of n g the energy levels are dominated by the charging part of the Hamiltonian. However, when n g is approximately half-integer and the charging energies of two adjacent states are close to each other (e.g., at V g V deg e/c g ), the Josephson tunneling mixes them strongly (see Fig. ). We concentrate on such a voltage range near a degeneracy point where only two charge states, say n0 and n1, play a role, while all other charge states, having a much higher energy, can be ignored. In this case the superconducting charge box (.1) reduces to a two-state quantum system (qubit) with a Hamiltonian that can be written in spin- 1 notation as H ctrl 1 B zˆ z 1 B xˆ x. (.3) The charge states n0 and n1 correspond to the spin basis states ( 1 0 ) and ( 0 1 ), respectively. The charging energy splitting, which is controlled by the gate voltage, corresponds in spin notation to the z component of the magnetic field, B z E ch 4E C 1n g, (.4) 3 In the ground state the superconducting state is totally paired, which requires an even number of electrons on the island. A state with an odd number of electrons necessarily costs an extra quasiparticle energy and is exponentially suppressed at low T. This parity effect has been established in experiments below a crossover temperature T* /(k B ln N eff ), where N eff is the number of electrons in the system near the Fermi energy (Tuominen et al., 199; Lafarge et al., 1993; Schön and Zaikin, 1994; Tinkham, 1996). For a small island, T* is typically one order of magnitude lower than the superconducting transition temperature. FIG.. The charging energy of a superconducting electron box is shown as a function of the gate charge n g for different numbers of extra Cooper pairs n on the island (dashed parabolas). Near degeneracy points the weaker Josephson coupling mixes the charge states and modifies the energy of the eigenstates (solid lines). In the vicinity of these points the system effectively reduces to a two-state quantum system. while the Josephson energy provides the x component of the effective magnetic field, B x E J. (.5) For later convenience we rewrite the Hamiltonian as H ctrl 1 Ecos zsin x, (.6) where the mixing angle tan 1 (B x /B z ) determines the direction of the effective magnetic field in the x-z plane, and the energy splitting between the eigenstates is E()B x B z E J /sin. At the degeneracy point, /, it reduces to E J. The eigenstates are denoted in the following as 0 and 1. They depend on the gate charge n g as 0cos sin, 1sin cos. (.7) We can further express the Hamiltonian in the basis of eigenstates. To avoid confusion we introduce a second set of Pauli matrices that operate in the basis 0 and 1, while reserving the operators for the basis of charge states and. By definition the Hamiltonian then becomes H 1 E z. (.8) The Hamiltonian (.3) is similar to the ideal singlequbit model (A1) presented in Appendix A. Ideally the bias energy (the effective magnetic field in the z direction) and the tunneling amplitude (the field in the x direction) are controllable. However, at this stage we can control by the gate voltage only the bias energy, while the tunneling amplitude has a constant value set by the Josephson energy. Nevertheless, by switching the gate voltage we can perform the required one-bit operations (Shnirman et al., 1997). If, for example, one chooses the idle state far to the left from the degeneracy point, the eigenstates 0 and 1 are close to and

5 Makhlin, Schön, and Shnirman: Quantum-state engineering 361, respectively. Then switching the system suddenly to the degeneracy point for a time t and back produces a rotation in spin space, U 1-bit exp i cos x i sin i sin cos, (.9) where E J t/. Depending on the value of t, a spin flip can be produced or, starting from 0, a superposition of states with any chosen weights can be reached. [This is exactly the operation performed in the experiments of Nakamura et al., (1999); see Sec. II.D.] Similarly, a phase shift between the two logical states can be achieved by changing the gate voltage n g for some time by a small amount, which modifies the energy difference between the ground and excited states. Several remarks are in order: (1) Unitary rotations by B x and B z are sufficient for all manipulations of a single qubit. By using a sequence of no more than three such elementary rotations we can achieve any unitary transformation of a qubit s state. () The example presented above, with control of B z only, provides an approximate spin flip for the situation in which the idle point is far from degeneracy and E C E J. But a spin flip in the logical basis can also be performed exactly. We must switch from the idle point idle to the point where the effective magnetic field is orthogonal to the idle one, idle /. This changes the Hamiltonian from H 1 E( idle ) z to H 1 E( idle /) x. To achieve this, the dimensionless gate charge n g should be increased by E J /(4E C sin idle ). For the limit discussed above, idle 1, this operating point is close to the degeneracy point, /. (3) An alternative way to manipulate the qubit is to use resonant pulses, i.e., ac pulses with frequency close to the qubit s level spacing. We do not describe this technique here as it is well known from NMR methods. (4) So far we have been concerned with the time dependence during elementary rotations. However, frequently the quantum state should be kept unchanged for some time, for instance, while other qubits are manipulated. Even in the idle state, idle, because the energies of the two eigenstates differ, their phases evolve relative to each other. This leads to coherent oscillations, typical for a quantum system in a superposition of eigenstates. We have to keep track of this time dependence with high precision and, hence, of the time t 0 from the very beginning of the manipulations. The timedependent phase factors can be removed from the eigenstates if all the calculations are performed in the interaction representation, with the zero-order Hamiltonian being the one at the idle point. However, the price for this simplification is an additional FIG. 3. A charge qubit with tunable effective Josephson coupling. The single Josephson junction is replaced by a fluxthreaded SQUID. The flux in turn can be controlled by a current-carrying loop placed on top of the structure. time dependence in the Hamiltonian during operations, introduced by the transformation to the interaction representation. This point has been discussed in more detail by Makhlin et al. (000b). (5) The choice of the qubit s logical basis is by no means unique. As follows from the preceding discussion, we can perform x and z rotations in the charge basis,,, which provides sufficient tools for any unitary operation. On the other hand, since we can perform any unitary transformation, we can choose any other basis as a logical basis as well. The Hamiltonian at the idle point is diagonal in the eigenbasis (.7), while the controllable part of the Hamiltonian, the charging energy, favors the charge basis. The preparation procedure (thermal relaxation at the idle point) is more easily described in the eigenbasis, while coupling to the meter (see Sec. V) is diagonal in the charge basis. So the choice of the logical states remains a matter of convention. (6) A final comment concerns normal-metal singleelectron systems. While they may serve as classical bits and logic devices, they are ruled out as potential quantum logic devices. The reason is that, due to the large number of electron states involved, their phase coherence is destroyed in the typical sequential tunneling processes. B. Charge qubit with tunable coupling A further step towards the ideal model (A1), in which the tunneling amplitude (x component of the field) is controlled as well, is the ability to tune the Josephson coupling. This is achieved by the design shown in Fig. 3, where the single Josephson junction is replaced by two junctions in a loop configuration (Makhlin et al., 1999). This dc SQUID is biased by an external flux x, which is coupled into the system through an inductor loop. If the self-inductance of the SQUID loop is low (Tinkham,

6 36 Makhlin, Schön, and Shnirman: Quantum-state engineering 1996), the SQUID-controlled qubit is described by a Hamiltonian of the form (.1) with modified potential energy: E J 0 cos x 0 E J 0 cos x 0 E 0 J cos x 0 cos. (.10) Here 0 hc/e denotes the flux quantum. We assume that the two junctions are identical 4 with the same E 0 J. The effective junction capacitance is the sum of individual capacitances of two junctions, in symmetric cases C J C 0 J. When parameters are chosen such that only two charge states play a role, we arrive again at the Hamiltonian (.3), but the effective Josephson coupling, B x E J x E 0 J cos x 0, (.11) is tunable. Varying the external flux x by amounts of order 0 changes the coupling between E 0 J and zero. 5 The SQUID-controlled qubit is thus described by the ideal single-bit Hamiltonian (A1), with field components B z (t)e ch V g (t) and B x (t)e J x (t) controlled independently by the gate voltage and the flux. If we fix in the idle state V g V deg and x 0 /, the Hamiltonian is zero, H 0 qb 0, and the state does not evolve in time. Hence there is no need to control the total time from the beginning of the manipulations, t 0. If we change the voltage or the current, the modified Hamiltonian generates rotations around the z or x axis, which are elementary one-bit operations. Typical time spans of single-qubit logic gates are determined by the corresponding energy scales and are of order /E J, /E ch for x and z rotations, respectively. If at all times at most one of the fields, B z (t) orb x (t), is turned on, only the time integrals of their profiles determine the results of the individual operations. Hence these profiles can be chosen freely to optimize the speed and simplicity of the manipulations. The introduction of the SQUID not only permits simpler and more accurate single-bit manipulations, but also allows us to control the two-bit couplings, as we shall discuss next. Furthermore, it simplifies the measurement procedure, which is more accurate at E J 0 (see Sec. V). C. Controlled interqubit coupling 4 While this cannot be guaranteed with high precision in an experiment, we note that the effective Josephson coupling can be tuned to zero exactly by a design with three junctions. 5 If the SQUID inductance is not small, the fluctuations of the flux within the SQUID renormalize the energy (.10). But still, by symmetry arguments, at x 0 / the effective Josephson coupling vanishes. FIG. 4. A register of many charge qubits coupled by oscillator modes in the LC circuit formed by the inductor and the qubit capacitors. In order to perform two-qubit logic gates we need to couple pairs of qubits together and to control the interactions between them. One possibility is to connect the superconducting boxes (i and j) directly, e.g., via a capacitor. The resulting charge-charge interaction is described by a Hamiltonian of the form (A) with an Isingtype coupling term z i z j. Such a coupling allows easy realization of a controlled-not operation. On the other hand, it has severe drawbacks. In order to control the two-bit interaction, while preserving the single-bit properties discussed above, one needs a switch to turn the two-bit interaction on and off. Any externally operated switch, however, connects the system to the dissipative external circuit, thus introducing dephasing effects (see Sec. IV). They are particularly strong if the switch is attached directly to the qubit and unscreened, which would be required in order to control the direct capacitive interaction. Therefore alternatives were explored in which the control fields were coupled only weakly to the qubits. A solution (Makhlin et al., 1999) is shown in Fig. 4. All N qubits are connected in parallel to a common LC-oscillator mode that provides the necessary two-bit interactions. It turns out that the ability to control the Josephson couplings by an applied flux simultaneously allows us to switch the two-bit interaction for each pair of qubits. This brings us close to the ideal model (A) with a coupling term y i y j. In order to demonstrate the mentioned properties of the coupling we consider the Hamiltonian of the chain (register of qubits) shown in Fig. 4: N H i1 en ic g V gi C J C g 1 NC qb q C qb C J i E J xi cos i en i L. (.1) Here q denotes the total charge accumulated on the gate capacitors of the array of qubits. Its conjugate variable is the phase drop across the inductor, related to the flux by // 0. Furthermore, C qb C JC g C J C g (.13)

7 Makhlin, Schön, and Shnirman: Quantum-state engineering 363 is the capacitance of the qubit in the external circuit. Depending on the relations among the parameters, the Hamiltonian (.1) can be reduced. We first consider the situation in which the frequency of the (q,) oscillator, (N) LC 1/NC qb L, is higher than typical frequencies of the qubit s dynamics: (N) LC E J,E ch. (.14) In this case the oscillator modes are not excited, but still their virtual excitation produces an effective coupling between the qubits. To demonstrate this we eliminate the variables q and and derive an effective description in terms of the qubits variables only. As a first step we perform a canonical transformation, q q(c qb / C J ) en i and i i (C qb /C J )(/ 0 ), while and n i are unchanged. This step leads to the new Hamiltonian (we omit the tildes) H q NC qb L i en ic g V gi C J C g E J xi cos i C qb. (.15) 0 C J We assume that the fluctuations of are weak, C qb C 0, (.16) J since otherwise the Josephson tunneling terms in the Hamiltonian (.15) are washed out (Shnirman et al., 1997). Assuming Eq. (.16) to be satisfied, we expand the Josephson terms in Eq. (.15) up to linear terms in. Then we can trace over the variables q and. Asa result we obtain an effective Hamiltonian, consisting of a sum of N one-bit Hamiltonians (.1) and the coupling terms C qb C J i H coup L 0 In spin- 1 notation this becomes 6 H coup ij where we introduced the scale E L E J xi sin i. (.17) E J xi E J xj E L ˆ yi ˆ yj const, (.18) C J C qb 0 L. (.19) The coupling Hamiltonian (.18) can be understood as the magnetic free energy of the current-biased inductor LI /. This current is the sum of the contributions from the qubits with nonzero Josephson coupling, I i E J i ( xi )sin i i E J i ( xi )ˆ yi. Note that the strength of the interaction does not depend directly on the number of qubits N in the system. 6 While expression (.18) is valid only in leading order in an expansion in E i J / N LC, higher terms also vanish when the Josephson couplings are put to zero. Hence the decoupling in the idle periods persists. (N) However, the frequency of the (q,) oscillator LC scales as 1/N. The requirement that this frequency not drop below typical eigenenergies of the qubit ultimately limits the number of qubits that can be coupled by a single inductor. A system with flux-controlled Josephson couplings E J ( xi ) and an interaction of the form (.18) allows us to perform all necessary gate operations in a straightforward way. In the idle state all Josephson couplings are turned off and the interaction (.18) is zero. Depending on the choice of idle state we may also tune the qubits by their gate voltages to the degeneracy points, which makes the Hamiltonian vanish, H0. The interaction Hamiltonian remains zero during one-bit operations, as long as we perform only one such operation at a time, i.e., for one qubit we have E i J E J ( xi )0. To perform i a two-bit operation for any pair of qubits, say, i and j, E J j and E J are switched on simultaneously, yielding the Hamiltonian H E J i ˆ x i E j J ˆ x j E J i j E J ˆ yi ˆ. (.0) E yj L While Eq. (.0) is not identical to Eq. (A) it equally well allows the relevant nontrivial two-bit operations, which, combined with the one-bit operations discussed above, provide a universal set of gates. A few comments should be added: (1) We note that typical time spans of two-bit operations are of the order E L /E J. It follows from conditions (.14) and (.16) that the interaction energy is never much larger than E J. Hence at best the two-bit gate can be as fast as a single-bit operation. () It may be difficult to fabricate a nanometer-scale inductor with the required inductance L, in particular, since it is not supposed to introduce stray capacitances. However, it is possible to realize such an element by a Josephson junction in the classical regime (with negligible charging energy) or an array of junctions. (3) The design presented above does not permit performing single- or two-bit operations simultaneously on different qubits. However, this becomes possible in more complicated designs in which parts of the many-qubit register are separated, for example, by switchable SQUID s. (4) In the derivation of the qubit interaction presented here we have assumed a dissipation-less highfrequency oscillator mode. To minimize dissipation effects, the circuit, including the inductor, should be made of superconducting material. Even so, at finite frequencies some dissipation will arise. To estimate its influence, the effect of Ohmic resistance R in the circuit has been analyzed by Shnirman et al. (1997), with the result that the interqubit coupling persists if the oscillator is underdamped, RL/NC qb. In addition the dissipation causes dephasing. An estimate of the resulting dephasing time can be obtained along the lines of the discussion in Sec. IV. For a

8 364 Makhlin, Schön, and Shnirman: Quantum-state engineering reasonably low-loss circuit the dephasing due to the coupling circuit is weaker than the influence of the external control circuit. (5) The interaction energy (.18) involves via E L the ratio of C J and C qb. The latter effectively screens the qubit from electromagnetic fluctuations in the voltage source s circuit, and hence should be taken as low as possible (see Sec. IV). Consequently, to achieve a reasonably high interaction strength and hence speed for two-bit operations, a large inductance is needed. For typical values of E J 100 mk and C g /C J 0.1 one needs an inductance of L 1 H in order not to have the two-bit operation more than ten times slower than the single-bit operation. However, large values of the inductance are difficult to reach without introducing large stray capacitances. To overcome this problem Makhlin et al. (000a) suggested using separate gate capacitors to couple the qubits to the inductor, as shown in Fig. 5. As long as the superconducting circuit of the inductor is at most weakly dissipative, there is no need to screen the qubit from the electromagnetic fluctuations in this circuit, and one can choose C L as large as C J (still larger C L would decrease the charging energy E C of the superconducting box), which makes the relevant capacitance ratio in Eq. (.17) of order one. Hence a fairly low inductance induces an interaction of sufficient strength. For instance, for the circuit parameters mentioned above, L10 nh would suffice. At the same time, potentially dephasing voltage fluctuations are screened by C g C J. (6) So far we have discussed manipulations on time scales much slower than the eigenfrequency of the LC circuit, which leave the LC oscillator permanently in the ground state. Another possibility is to use the oscillator as a bus mode, in analogy to the techniques used for ion traps. In this case an ac voltage with properly chosen frequency is applied to a qubit to entangle its quantum state with that of the LC circuit (for instance, by exciting the oscillator conditionally on the qubit s state). Then by addressing another qubit one can absorb the oscillator quantum, simultaneously exciting the second qubit. As a result, a two-qubit unitary operation is performed. This coupling via real excitations is a firstorder process, as opposed to the second-order interaction (.18). Hence this method allows for faster two-qubit operations. Apart from this technical advantage, the creation of entanglement between a qubit and an oscillator would by itself be a very interesting experimental achievement (Buisson and Hekking, 000). FIG. 5. A register of charge qubits coupled to an inductor via separate capacitors C L C J, independent from the gate capacitors C g. D. Experiments with Josephson charge qubits Several of the concepts and properties described above have been verified in experiments. This includes the demonstration of superpositions of charge states, the spectroscopic verification of the spectrum, and even the demonstration of coherent oscillations. In a superconducting charge box the coherent tunneling of Cooper pairs produces eigenstates that are gatevoltage-dependent superpositions of charge states. This property was first observed, in a somewhat indirect way, in the dissipative current through superconducting single-electron transistors. In this system single-electron tunneling processes (typically treated in perturbation theory) lead to transitions between the eigenstates. Since the eigenstates are not pure charge states, the Cooper-pair charge may also change in a transition. In the resulting combination of coherent Cooper-pair tunneling and stochastic single-electron tunneling the charge transferred is not simply e and the work done by the voltage source not simply ev. [In an expansion in the Josephson coupling to nth order the charge (n 1)e is transferred.] As a result a dissipative current can be transferred at subgap voltages. The theoretical analysis predicted a richly structured I-V characteristic at subgap voltages (Averin and Aleshkin, 1989; Maassen van den Brink et al., 1991; Siewert and Schön, 1996), which has been qualitatively confirmed by experiments (Maassen van den Brink et al., 1991; Tuominen et al., 199; Hadley et al., 1998). A more direct demonstration of eigenstates that arise as superpositions of charge states was found in the Saclay experiments (Bouchiat, 1997; Bouchiat et al., 1998). In their setup (see Fig. 6) a single-electron transistor was coupled to a superconducting charge box (as in the measurement setup to be discussed in Sec. V) and the expectation value of the charge of the box was measured. When the gate voltage was increased adiabatically this expectation value increased in a series of rounded steps near half-integer values of n g. At low temperatures the width of this transition agreed quantitatively with the predicted ground-state properties of Eqs. (.3) and (.7). At higher temperatures, the excited state contributed, again as expected from theory. Next we mention the experiments of Nakamura et al. (1997), who studied the superconducting charge box by spectroscopic means. When exposing the system to radiation they found resonances (in the tunneling current in a suitable setup) at frequencies corresponding to the

9 Makhlin, Schön, and Shnirman: Quantum-state engineering 365 FIG. 6. Scanning electron micrograph of a Cooper-pair box coupled to a single-electron transistor used in the experiments of the Saclay group (Bouchiat, 1997; Bouchiat et al., 1998). difference in the energy between excited and ground states, again in quantitative agreement with the predictions of Eq. (.3). The most spectacular demonstration so far of the concepts of Josephson qubits has been provided by Nakamura et al. (1999). Their setup is shown in Fig. 7. In these experiments the Josephson charge qubit was prepared far from the degeneracy point for a sufficiently long time to relax to the ground state. In this regime the ground state was close to a charge state, say,. Then the gate voltage was suddenly switched to a different value. Let us first discuss the case in which it was switched precisely to the degeneracy point. Then the initial state, a pure charge state, was an equal-amplitude superposition of the ground state 0 and the excited state 1. These two eigenstates have different energies, hence in time they acquire different phase factors: te ie 0 t/ 0e ie 1 t/ 1. (.1) After a delay time t the gate voltage was switched back to the original gate voltage. Depending on the delay, the system then ended up either in the ground state [for (E 1 E 0 )t/h with integer], in the excited state [for (E 1 E 0 )t/h(1)], or in general in a t-dependent superposition. The probability that, as a result of this manipulation, the qubit is in the excited state is measured by monitoring the current through a probe junction. In the experiments this current was averaged over many repeated cycles, involving relaxation and switching processes, and the oscillatory dependence on t described above was observed. In fact even more details of the theory have been quantitatively confirmed. For instance, one also expects and finds an oscillatory behavior when the gate voltage is switched to a point different from the degeneracy point, with the frequency of oscillations being a function of this gate voltage. Second, the frequency of the coherent oscillations depends on the Josephson coupling energy. The latter can be varied, since the Josephson coupling is controlled by a flux-threaded SQUID (see Fig. 3). This aspect has also been verified quantitatively. Coherent oscillations with a period of roughly 100 ps could be observed in the experiments of Nakamura et al. (1999) for at least ns. 7 This puts a lower limit on the phase coherence time and, in fact, represents its first direct measurement in the time domain. Estimates show that a major contribution to the dephasing is the measurement process by the probe junction itself. In the experiments so far the detector was permanently coupled to the qubit and observed it continuously. Still, information about the quantum dynamics could be obtained since the coupling strength was optimized: it was weak enough not to destroy the quantum time evolution too quickly and strong enough to produce a sufficient signal. A detector that does not induce dephasing during manipulations should significantly improve the operation of the device. In Sec. V we suggest using a single-electron transistor, which performs a quantum measurement only when switched to a dissipative state. So far only experiments with single qubits have been sucessfully carried out. Obviously the next step is to couple two qubits and to create and detect entangled states. Experiments in this direction have not yet been successful, partially because of difficulties such as, for instance, dephasing due to fluctuating background charges. However, the experiments using single qubits imply that extensions to coupled qubits should be possible as well. E. Adiabatic charge manipulations Another qubit design, based on charge degrees of freedom in Josephson-junction systems, was proposed by Averin (1998). It also allows control of two-bit coupling at the price of representing each qubit by a chain of Josephson-coupled islands. The basic setup is shown in Fig. 8. Each superconducting island (with index i) is FIG. 7. Micrograph of a Cooper-pair box with a fluxcontrolled Josephson junction and a probe junction (Nakamura et al., 1999). 7 In later experiments the same group reported phase coherence times as long as 5 ns (Nakamura et al., 000).

10 366 Makhlin, Schön, and Shnirman: Quantum-state engineering FIG. 8. Two coupled qubits as proposed by Averin (1998). biased via its own gate capacitor by a gate voltage V i. The control of these voltages allows one to move the charges along the chain analogously to the adiabatic pumping of charges in junction arrays (see, for example, Pekola et al., 1999). The capacitances of the Josephson junctions as well as the gate capacitances are small enough so that the typical charging energy prevails over the Josephson coupling. In this regime the appropriate basis is that of charge states n 1,n,..., where n i is the number of extra Cooper pairs on island i. There exist gate voltage configurations such that the two charge states with the lowest energy are almost degenerate, while all other charge states have much higher energy. For instance, if all voltages are equal except for the voltages V m and V l at two sites, m and l, one can achieve a situation in which the states 0,0,0,... and 0,...,1 m,0,...,1 l,... are degenerate. The subspace of these two charge states is used as the logical Hilbert space of the qubit. They are coupled via Josephson tunneling across the ml1 intermediate junctions. The parameters of the qubits Hamiltonian can be tuned via the bias voltages. Obviously the bias energy B z (V 1,V,...) between these two states can be changed via the local voltages V l and V m. Furthermore, the effective tunneling amplitude B x (V 1,V,...) can be tuned by adiabatic pumping of charges along the chain, changing their distance ml and hence the effective Josephson coupling, which depends exponentially on this distance. (The Cooper pair must tunnel via ml1 virtual charge states with much higher energy.) An interqubit interaction can be produced by placing a capacitor between the edges (outer islands) of two qubits. If at least one of the charges in each qubit is shifted closer to this capacitor, the Coulomb interaction leads to an interaction of the type J zz z 1 z. The resulting two-bit Hamiltonian is of the form H 1 B j z t j z B j x t j x J zz t 1 z z. j1, (.) For controlled manipulations of the qubit the coefficients of the Hamiltonian are modified by adiabatic motion of the charges along the junction array. The adiabaticity is required to suppress transitions between different eigenstates of the qubit system. While conceptually satisfying, this proposal appears difficult to implement: It requires many gate voltages for each qubit. Due to the complexity a high degree of accuracy is required for the operation. Its larger size as compared to simpler designs makes the system more vulnerable to dephasing effects, for example, due to fluctuations of the offset charges. Adiabatic manipulations of the Josephson charge qubit can lead to Berry phases. Falci et al. (000) suggested that a Berry phase could accumulate during suitable manipulations of a flux-controlled charge qubit with an asymmetric dc SQUID, and that it could be detected in an experiment similar to that of Nakamura et al. (1999). If the bare Josephson couplings of the SQUID loop are E J 1 and E J the effective Josephson energy is given by [cf. Eq. (.10)] E J 1 cos x 0 E J cos x 0. (.3) Hence the corresponding Hamiltonian of the qubit has all three components of the effective magnetic field: B x (E 1 J E J )cos( x / 0 ) and B y (E J E 1 J )sin( x / 0 ), while B z is given by Eq. (.4). With three nonzero field components, adiabatic changes of the control parameters V g and x may result in B s enclosing a nonzero solid angle. This results in a Berry phase shift B between the ground and excited states. In general, a dynamic phase E(t)dt is also accumulated in the process. To single out the Berry phase, Falci et al. (000) suggested encircling the loop in parameter space back and forth, with a NOT operation performed in between. The latter exchanges the ground and excited states, and as a result the dynamic phases accumulated during both paths cancel. At the same time the Berry phases add up to B. This phase shift can be measured by a procedure similar to that used by Nakamura et al. (1999): the system is prepared in a charge state away from degeneracy, abruptly switched to the degeneracy point where adiabatic manipulations and the NOT gate are performed, and then switched back. Finally, the average charge is measured. The probability of finding the qubit in the excited charge state sin B reflects the Berry phase. The experimental demonstration of topological phases in Josephson-junction devices would constitute a new class of macroscopic quantum effects in these systems. They could be performed with a single Josephson qubit in a design similar to that used by Nakamura et al. (1999) and thus appear feasible in the near future. III. QUBITS BASED ON THE FLUX DEGREE OF FREEDOM In the previous section we described the quantum dynamics of low-capacitance Josephson devices where the charging energy dominates over the Josephson energy, E C E J, and the relevant quantum degree of freedom is the charge on superconducting islands. We shall now review the quantum properties of superconducting devices in the opposite regime, E J E C, where the flux is the appropriate quantum degree of freedom. These systems were proposed by Caldeira and Leggett (1983) in the mid 1980s as test objects to study various quantummechanical effects, including macroscopic quantum tunneling of the phase (or flux) as well as resonance tunneling. Both had been observed in several experiments

11 Makhlin, Schön, and Shnirman: Quantum-state engineering 367 FIG. 9. The simplest flux qubits: (a) The rf SQUID, a simple loop with a Josephson junction, forms the simplest Josephson flux qubit; (b) improved design for a flux qubit. The flux x in the smaller loop controls the effective Josephson coupling of the rf SQUID. (Voss and Webb, 1981; Martinis et al., 1987; Clarke et al., 1988; Rouse et al., 1985). Another important quantum effect has been reported recently: The groups at Stony Brook (Friedman et al., 000) and Delft (van der Wal et al., 000) have observed in experiments the avoided level crossing due to coherent tunneling of the flux in a double-well potential. In principle, all other manipulations discussed in the previous section should be possible with Josephson flux devices as well. They have the added advantage of not being sensitive to fluctuations in the background charges. However, attempts to observe macroscopic quantum coherent oscillations in Josephson flux devices have not been yet successful (Leggett, 1987; Tesche, 1990). A. Josephson flux (persistent current) qubits We consider superconducting ring geometries interrupted by one or several Josephson junctions. In these systems persistent currents flow and magnetic fluxes are enclosed. The simplest design of these devices is an rf SQUID, which is formed by a loop with one junction, as shown in Fig. 9(a). The phase difference across the junction is related to the flux in the loop (in units of the flux quantum 0 h/e) by // 0 integer. An externally applied flux x biases the system. Its Hamiltonian, with Josephson coupling, charging energy, and magnetic contributions taken into account, thus reads HE J cos 0 x Q. (3.1) L C J Here L is the self-inductance of the loop and C J the capacitance of the junction. The charge Qi/ on the leads is canonically conjugate to the flux. If the self-inductance is large, such that L E J /( 0 /4 L) is larger than 1 and the externally applied flux x is close to 0 /, the first two terms in the Hamiltonian form a double-well potential near 0 /. At low temperatures only the lowest states in the two wells contribute. Hence the reduced Hamiltonian of this effective two-state system again has the form (.3), H ctrl 1 B z ˆ z 1 B x ˆ x. The diagonal term B z is the bias, i.e., the asymmetry of the double-well potential, given for L 11 by B z x 46 L 1 E J x / 0 1/. (3.) B z can be tuned by the applied flux x. The offdiagonal term B x describes the tunneling amplitude between the wells, which depends on the height of the barrier and thus on E J. This Josephson energy, in turn, can be controlled if the junction is replaced by a dc SQUID, as shown in Fig. 9(b), introducing the flux x as another control variable. 8 With these two external control parameters the elementary single-bit operations, i.e., z and x rotations, can be performed, equivalent to the manipulations described for charge qubits in the previous section. In addition, for flux qubits we can either perform the operations by sudden switching of the external fluxes x and x for a finite time, or we can use ac fields and resonant pulses. To permit coherent manipulations the parameter L should be chosen larger than unity (so that two wells with well-defined levels appear) but not much larger, since the resulting large separation of the wells would suppress the tunneling. The rf SQUID described above had been discussed in the mid 1980s as a realization of a two-state quantum system. Some features of macroscopic quantum behavior were demonstrated, such as macroscopic quantum tunneling of the flux, resonant tunneling, and level quantization (Voss and Webb, 1981; Martinis et al., 1987; Clarke et al., 1988; Rouse et al., 1995; Silvestrini et al., 1997). However, only very recently has the level repulsion near a degeneracy point been demonstrated (Friedman et al., 000; van der Wal et al., 000). The group at Stony Brook (Friedman et al., 000) probed spectroscopically the superposition of excited states in different wells. The rf SQUID used had selfinductance L40 ph and L.33. A substantial separation of the minima of the double-well potential (of order 0 ) and a high interwell barrier made the tunnel coupling between the lowest states in the wells negligible. However, both wells contain a set of higher localized levels under suitable conditions one state in each well with relative energies also controlled by x and x. Because they were closer to the top of the barrier, these states mixed more strongly and formed eigenstates, which were superpositions of localized flux states from different wells. External microwave radiation was used to pump the system from a well-localized lowest state in one well to one of these eigenstates. The energy spectrum of these levels was studied for different biases x, x, and the properties of the model (3.1) were confirmed. In particular, the level splitting at the degeneracy point indicated a superposition of distinct quantum states. They differed in a macroscopic way: the authors estimated that the two superimposed flux states differed in flux by 0 /4, in current by 3 A, and in magnetic moment by B. 8 See Mooij et al. (1999) for suggestions on how to control x independent of x.

12 368 Makhlin, Schön, and Shnirman: Quantum-state engineering The Delft group (van der Wal et al., 000) performed microwave spectroscopy experiments on a similar but much smaller three-junction system, to be described in more detail below. In their system the superpositions of the lowest states in two wells of the Josephson potential landscape were probed. In the spectrum they observed a level repulsion at the degeneracy point, confirming the predictions of the two-state model Hamiltonian (.3) with the parameters B x, B z calculated from the potential (3.3). In spite of this progress, attempts to observe macroscopic quantum coherence, i.e., the coherent oscillations of a quantum system prepared in a superposition of eigenstates, have not been successful so far (Leggett, 1987; Tesche, 1990). A possible reason for this failure was suggested recently by Mooij et al. (1999). They argue that for the designs considered so far the existence of the double-well potential requires that L 1, which translates into a rather high product of the critical current of the junction and its self-inductance. In practice, only a narrow range of circuit parameters is useful, since high critical currents require a relatively large junction area resulting in a high capacitance, which suppresses tunneling. A high self-inductance of the rf SQUID can be achieved only in large loops. This makes the system very susceptible to external noise. To overcome this difficulty Mooij et al. (1999) and Feigelman et al. (000) proposed using a smaller superconducting loop with three or four junctions, respectively. Here we discuss the three-junction circuit shown in Figs. 10(a) and (c). In this low-inductance circuit the flux through the loop remains close to the externally applied value, x. Hence the phase differences across the junctions are constrained by 1 3 x / 0, leaving 1 and as independent dynamical variables. In the plane spanned by these two variables the Josephson couplings produce a potential landscape given by U 1, E J cos 1 E J cos Ẽ J cos x / 0 1. (3.3) If Ẽ J /E J 0.5, a double-well potential is formed within each cell in the phase plane. For an optimal value of Ẽ J /E J the cells are separated by high barriers, while tunneling between two minima within one cell is still possible. The lowest states in the wells form a two-state quantum system, with two different current configurations. Mooij et al. (1999) and Orlando et al. (1999) discuss junctions with E J K and E J /E C 80 and loops of micrometer size with very small selfinductance L5 ph (which can be neglected when calculating the energy levels). Typical qubit operation parameters are the level splitting B z 0.5 K and the tunneling amplitude B x 50 mk. For the optimal choice of Ẽ J /E J the two minima differ in phases by an amount of order /. Due to the very low inductance and the relatively low critical current I c 00 na, this translates FIG. 10. The Delft design of a flux qubit: (a) and (c) A threejunction loop as a flux qubit (Mooij et al., 1999). The reduced size and lower inductance of this system as compared with earlier designs [e.g., Fig. 9(a)] reduce the coupling to the external world and hence dephasing effects. (b) Multijunction flux qubit with a controlled Josephson coupling (Mooij et al., 1999). Control over two magnetic fluxes, and, allows one to perform all single-qubit logic operations. into a flux difference of LI c While this corresponds to a still macroscopic magnetic moment of B, the two basis states are similar enough to make the coupling to external fluctuating fields and hence the dephasing effects weak (for a further discussion, see Sec. IV). In this respect the new design is qualitatively superior to the simple rf SQUID. In order to obtain more direct evidence for superpositions of localized flux states the Delft group (van der Wal et al., 000) measured the average flux of the qubit as a function of the external bias B z. This experiment is similar to that of Bouchiat et al. (1997, 1998) for the single-cooper-pair box, which was discussed in Sec. II.D. As the bias is swept across the degeneracy point, one expects the average flux to change from the value in one well to that in the other well. At high temperatures the step is rounded, with width set by temperature. As k B T is lowered below the tunnel splitting B x this width should saturate at the value of B x. However, experimentally it saturated much earlier than expected from the spectroscopically measured tunnel splitting. This discrepancy indicates an enhanced population of the excited state, which could be caused by noise, either from external sources or due to the dc-squid detector.

13 Makhlin, Schön, and Shnirman: Quantum-state engineering 369 FIG. 11. Flux qubits coupled in two ways. The dashed line induces a direct inductive coupling. Alternatively, an interqubit coupling is provided by the LC circuit indicated by a solid line. B. Coupling of flux qubits In order to couple different flux qubits one can use a direct inductive coupling (Mooij et al., 1999; Orlando et al., 1999), as shown by the dashed line in Fig. 11. A mutual inductance between the qubits can be established in different ways. The dashed loop shown in the figure couples the currents and fluxes in the lower parts of the qubits. Since fluxes through these loops control the barrier heights of the double-well potentials, this gives rise to the interaction term ˆ x1 ˆ x. Placing the loop differently produces in addition contributions to the interaction Hamiltonian of the form ˆ z1 ˆ z. The typical interaction energy is of order MI c where M is the mutual inductance and I c (/ 0 )E J is the critical current in the junctions. For their design, Mooij et al. (1999) estimate the typical interaction energy to be of order 0.01E J 50 mk in frequency units, i.e., of the order of single-qubit energies. For a typical rf SQUID (Friedman et al., 000) this coupling can be even stronger than the tunneling rate between the flux states of the SQUID. In the simplest form this interaction is always turned on. To turn it off completely, one needs a switch controlled by high-frequency pulses. The related coupling to the external circuit leads to decoherence (see the discussion at the end of this section). An alternative is to keep the interaction turned on constantly and use ac driving pulses to induce coherent transitions between the levels of the two-qubit system (see Shnirman et al., 1997; Mooij et al., 1999). A disadvantage of this approach is that permanent couplings result in an unwanted accumulation of relative phases between the two-qubit states even in the idle periods. Keeping track of these phases, or their suppression by repeated refocusing pulses (see Sec. IV), requires a high precision and complicates the operation. A controllable interqubit coupling without additional switches is achieved in the design shown by the solid line in Fig. 11 (Makhlin et al., 000c). The coupling is mediated by an LC circuit, with self-inductance L osc and capacitance C osc, which is coupled inductively to each qubit. Like the design of the charge qubit register in Sec. II.C, the coupling depends on parameters of individual qubits and can be controlled in this way. The effective coupling can be found again by integrating out the fast oscillations in the LC circuit. It can be understood in a simple way by noting that in the limit C osc 0 the qubits establish a voltage drop across the inductor, V i M i /L, and the Hamiltonian for the oscillator mode is H osc /L osc Q /C osc VQ, with the charge Q being conjugate to the flux through the LC circuit. Here i is the flux in the loop of qubit i, L is the selfinductance of the loop, and M is its mutual inductance with the LC circuit. Continuing as described in Sec. II.C, we obtain the interqubit interaction term C osc V /. In the limit of weak coupling to the LC circuit, we have i(i/)h i, i i B i x ˆ yi /, where i is the separation between two minima of the potential and B x is the tunneling amplitude. Hence the i interaction is given by H int M L ij i j 0 B x i B x j e /C osc ˆ yi ˆ yj. (3.4) To turn off the interaction one should suppress the tunneling amplitudes B i x. This can be done with exponential precision by increasing the height of the potential barrier via x. Note that in this case unwanted fluctuations of B x and resulting dephasing effects are also i exponentially suppressed. All needed single and twoqubit manipulations can be performed by turning on the i fields B x and B i z, in complete analogy to what we discussed in Sec. II.C. We also encounter the equivalent drawbacks: the design shown in Fig. 11 does not allow simultaneous manipulations on different qubit pairs, and the conditions of high oscillator frequencies and weak renormalization of qubit parameters by the coupling, similar to Eqs. (.14) and (.16), limit the two-qubit coupling energy. The optimization of this coupling requires L osc /C osc R K (/ 0 ) (M/L) and LC not far above the qubit frequencies. For rf SQUID s (Friedman et al., 000) the resulting coupling can reach the same order as the single-bit terms. On the other hand, for the design of Mooij et al. (1999), in which two basis phase states differ only slightly in their magnetic properties, the coupling term is much weaker than the single-bit energies. C. Quiet superconducting phase qubits The circuits considered so far in this section are vulnerable to external noise. First, they need for their operation an external bias in the vicinity of 0 /, which should be kept stable for the time of manipulations. In addition, the two basis flux states of the qubit have different current configurations, which may lead to magnetic interactions with the environment and possible cross talk between qubits. To a large extent the latter effect is suppressed already in the design of Mooij et al. (1999). To further reduce these problems several designs of so-called quiet qubits have been suggested (Ioffe et al., 1999; Zagoskin, 1999; Blais and Zagoskin, 000; Blatter, 001) They are based on intrinsically doubly degenerate systems, e.g., Josephson junctions with d-wave leads and energy-phase relations (e.g., cos ) with two minima, or the use of junctions, which removes the need for a constant magnetic bias near 0 /.

14 370 Makhlin, Schön, and Shnirman: Quantum-state engineering FIG. 1. A five-junction loop, a basic bistable element of quiet superconducting qubits (Blatter et al., 001), is made of four ordinary junctions and one stronger junction. In two stable configurations the phase difference across this element is /. The relevant two states differ only in their distribution of internal currents in the Josephson junctions while external loops carry no current. As a result the coupling of the qubit to the electromagnetic environment is substantially reduced and coherence is preserved longer. Designs have been proposed using tunnel junctions between s- and d-wave superconductors (SD), between two d-wave superconductors (DD), and between two d-wave superconductors with a normal-metal bridge (DND). The mentioned designs are similar and we discuss them in parallel. Ioffe et al. (1999) suggested using an SD tunnel junction with the s-wave lead matched to the (110) boundary of the d-wave superconductor. In this geometry the first harmonic, cos, in the Josephson coupling vanishes due to symmetry reasons, and one obtains a bistable system with the potential energy E J cos and minima at 0 with 0 /. [A similar current-phase relation was observed recently by II ichev et al. (1998) in a DD junction with a mismatch angle of 45.] The DND design, with different orientations for the two d-wave superconductors, was proposed by Zagoskin (1999). The energy-phase relation for such junctions also has two degenerate minima, at the phase differences 0. The separation 0 of these minima, and hence the tunneling amplitude, are controlled by the mismatch angle of the d-wave leads. In a later development a macroscopic analog of d-wave qubits was discussed (Blatter et al., 001). Instead of an SD junction, it is based on a five-junction loop, shown in Fig. 1, which contains one strong junction and four ordinary junctions. The presence of the junction is equivalent to magnetically biasing the loop with a half superconducting flux quantum. Four other junctions, frustrated by the phase shift, have two lowest-energy states with the phase difference of / between the external legs in the figure. In this respect the five-junction loop is similar to the SD junction discussed above and can be called a / junction. In all these designs the bistability is a consequence of time-reversal symmetry (which changes the signs of all the phases) of the Hamiltonian. Thus the degeneracy also persists in systems containing different Josephson junctions, although the phase differences in the two lowest-energy states and their separation can change. If charging effects with E C E J are included, one arrives at a double-well system with tunneling between the wells. Such a qubit can be operated by connecting or disconnecting it from external elements, as described below. The first issue to be addressed is how to store the qubit s state, i.e., how to freeze the evolution. This can be achieved by connecting the qubit in parallel to a large capacitor (Ioffe et al., 1999). This makes the phase degree of freedom very massive, thus suppressing the tunneling and restoring the needed degeneracy. In order to perform a ˆ x rotation, one turns on interwell tunneling by disconnecting the capacitor. This means a switch is needed in the circuit. The ˆ z rotation or phase shift can be accomplished by lifting the degeneracy between the wells. This can be done by connecting another, much stronger / junction (an SD-junction or five-junction loop) and a weak ordinary s-wave junction (with Josephson energy cos ) in series to the qubit, to form a closed loop. This again requires a switch. The auxiliary / junction shifts the phase differences of the potential minima of the qubit to 0 and. Hence the s junction is in the ground state or frustrated depending on the qubit s phase drop. The corresponding energy difference produces the needed phase shift between two qubit s states. To perform two-qubit manipulations and control the entanglement, Ioffe et al. (1999) proposed forming a loop, connecting in series two qubits and one s junction with weak Josephson coupling E J s E J. The phase state of each qubit is characterized by the phase difference of 0, i.e., the total phase drop on the qubits is equal to 0 or 0 depending on whether the qubits are in the same state or in different ones. When the connection between the qubits and the s junction is turned on, this phase drops across the s junction, and its energy differs by E J s (1cos 0 ) for the states 00, 11 as compared to the states 01, 10. The net effect is an Ising-type interaction between the pair of qubits, which allows unitary two-qubit transformations. Another mode of operation was discussed by Blais and Zagoskin (000). They suggested using a magnetic force microscope tip for single-bit manipulations (local magnetic field lifts the degeneracy of two phase states) and for the readout of the phase state. The tip should be moved towards or away from the qubit during manipulations. The short time scales of qubit operation make this proposal difficult to realize. Even in quiet designs, in both SD and DD systems, there are microscopic persistent currents flowing inside the junctions which differ for the two logical states (Zagoskin, 1999; Blatter et al., 001). These weak currents still couple to the outside world and to other qubits, thus spoiling the ideal behavior. Furthermore, all the designs mentioned require externally operated switches to connect and disconnect qubits. We discuss the associated problems in the following subsection. To summarize, the quiet designs require rather complicated manipulations as well as circuits with many junctions, including junctions or d-wave junctions, which are difficult to fabricate in a controlled and reliable way. In addition, many constraints imposed on the circuit parameters (in particular, on the hierarchy of Josephson couplings) appear difficult to satisfy. In our

15 Makhlin, Schön, and Shnirman: Quantum-state engineering 371 opinion the quiet phase qubit designs belong to a highercomplexity class than the previously discussed charge and flux qubits, and their experimental realization may remain a challenge for some time. D. Switches Switches may be used in a variety of contexts in quantum nanocircuits. They are needed, for example, for a direct capacitive coupling between charge qubits or magnetic coupling of flux qubits. They are also a major tool for controlling the dynamics of quiet qubits. Ideal switches should decouple qubits from the environment and at the same time let through control signals. They should operate on the very fast time scale of the qubit dynamics and have a high switching ratio, that is, the ratio of the interaction with the switch in the on or off state. Such switches are hard to realize. In this section we compare the characteristics of several Josephsonjunction-based switches and their associated problems. Possible switches are dc SQUID s as well as SSET s (superconducting single-electron transistors, or single- Cooper-pair transistors) in a mode in which they act as Josephson junctions with an externally controlled coupling. Then the switching ratio is the ratio of the minimal and maximal values of the coupling. In a dc SQUID with Josephson energies of its junctions equal to E J 1 and E J, this ratio is (E J 1 E J )/(E J 1 E J ). It reached a value below 1% in the experiment of Rouse et al. (1995). However, fast switching of the bias flux may be difficult to perform. In a SSET the effective coupling is controlled by a gate voltage, which can be switched quickly. However, the switching ratio of order E J /E C (E J and E C are characteristics of the SSET) is hardly below several percent. These limitations lead to unwanted interactions when the switch is supposed to be disconnected. Since a dc SQUID requires an external bias to be operated as a switch, Blatter et al. (001) suggested a similar construction with the bias provided by / junctions instead of an external magnetic field. That is, one could insert two / junctions into one arm of the SQUID loop. Depending on whether the phase drops across these junctions were equal or opposite, they would simulate an external bias of a half flux quantum or no bias. Accordingly, the Josephson couplings of two s junctions in the SQUID would add up or cancel each other. The switching could be realized via a voltage pulse that drives one / junction between / and / states. Blatter et al. (001) also suggested using an array of n such switches, reducing the overall Josephson coupling in the off state by a factor (E J 1 E J )/E C n. Unfortunately, in the on state the overall coupling through the array would also be reduced with growing n, although this reduction might be weaker than in the off state, i.e., the switching ratio increases with n. Nevertheless the quality of the switch in the on state would be reduced. Moreover, to operate the switch one would need to send voltage pulses simultaneously to n intermediate elements, which complicates the operation. Note that this design is reminiscent of the qubit design proposed by Averin (1998), which is presented in Sec. II.E. However, while Blatter et al. (001) suggest controling the coupling, (E J /E C ) n by controlling E J, Averin proposes to changing the distance n of the tunneling process. While switches of the type described above may be useful in first experiments with simple quantum nanocircuits, further work is needed before they can be used in more advanced designs that require high precision of manipulations and phase coherence over a long period of time. IV. ENVIRONMENT AND DISSIPATION A. Identifying the problem For an ideal quantum system the time evolution is described by deterministic, reversible unitary operations. The concepts of quantum-state engineering and computation heavily rely on this quantum coherence, with many potential applications requiring a large number of coherent manipulations of a large number of qubits. On the other hand, for any real physical quantum system the time evolution may be disturbed in various ways, and the number of coherent manipulations is limited. Possible sources of error are inaccuracies in the preparation of the initial state, inaccuracies in the manipulations (logic gates), uncontrolled couplings between qubits, undesired excitations out of the two-state Hilbert space (Fazio et al., 1999), and unavoidable in devices that are to be controlled externally interactions with the environment. Due to the coupling to the environment, the quantum state of the qubits gets entangled with the environmental degrees of freedom. As a consequence the phase coherence is destroyed after a time scale called the dephasing time. In this section we shall describe the influence of the environment on the qubit. We determine how the dephasing time depends on system parameters and how it can be optimized. Some of the errors can be corrected by software tools. One known from NMR and, in particular, NMR-based quantum logic operations (see, for example, Chang, 1998) is refocusing. Refocusing techniques serve to suppress the effects of undesired terms in the Hamiltonian, e.g., deviations of the single-bit field terms from their nominal values or uncontrolled interactions like stray direct capacitive couplings of charge qubits or inductive couplings of flux qubits. As an example we consider the error due to a single-bit term B x x, which after some time has produced an unwanted rotation by. Refocusing is based on the fact that a pulse about the z axis reverses the influence of this term, i.e., U z ()U x ()U z ()U x (). Hence fast repeated inversions of the bias B z (t) (with B z B x ) eliminate the effects associated with B x. The technique can also be applied to enhance the precision of nonideal control switches: one first turns off the coupling term to a low value and then further suppresses it by refocusing. The examples demonstrate that refocusing requires very fast

16 37 Makhlin, Schön, and Shnirman: Quantum-state engineering repeated switchings with a period much shorter than the elementary operation time. This can make it hard to implement. It was therefore a major breakthrough when the concepts of quantum error correction were discovered (see, for example, Preskill, 1998; Steane, 1998). When applied they should make it possible, even in the presence of dephasing processes provided that the dephasing time is not too short to perform coherent sequences of quantum manipulations of arbitrary length. The price to be paid is an increase in system size (by roughly an order of magnitude), and a large number of steps are needed for error correction before another computational step can be performed (increasing the number of steps by roughly three orders of magnitude). This imposes constraints on the dephasing time. Detailed analysis shows that error correction can be successful if the dephasing time is of the order of 10 4 times longer than the time needed for an elementary logic gate. In the Josephson-junction systems discussed here, the environment is usually composed of resistive elements in the circuits needed for the manipulations and the measurements. They produce voltage and current noise. In many cases such fluctuations are Gaussian distributed with a Johnson-Nyquist power spectrum, coupling linearly to the quantum system of interest. They can thus be described by a harmonic oscillator bath with suitable frequency spectrum and coupling strength (Leggett et al., 1987; Weiss, 1999). For charge qubits, for instance, fluctuations in the gate voltage circuit, coupling to z, and fluctuations in the current, which control the Josephson energy and couple to x, can be described in this way (Shnirman et al., 1997). In this section we shall first describe these noise sources and the dephasing then introduce. We later comment on other noise sources such as telegraph noise, typically with a 1/f power spectrum due to switching two-level systems (e.g., background charge fluctuations), or the shot noise resulting from tunneling in a single-electron transistor coupled to a qubit for the purpose of a measurement. Depending on the relation between typical frequencies of the coherent (Hamiltonian) dynamics and the dephasing rates, we distinguish two regimes. In the first, the Hamiltonian-dominated regime, where the controlled part of the qubit Hamiltonian H ctrl (1/)B, governing the deterministic time evolution and logic gates, is large, it is convenient to describe the dynamics in the eigenbasis of H ctrl. The coupling to the environment is weak, hence the environment-induced transitions are slow. One can then distinguish two stages: (a) dephasing processes, in which the relative phase between the eigenstates becomes random; and (b) energy relaxation processes, in which the occupation probabilities of the eigenstates change. In the second, environment-dominated, regime H ctrl is too weak to support its eigenstates as the preferred basis. The qubit s dynamics in this situation is governed by dissipative terms and depends on details of the structure of the coupling to the environment. In general the evolution is complicated, and the distinction between relaxation and dephasing may be impossible. Both regimes may be encountered during manipulations. Obviously the Hamiltonian should dominate when a coherent manipulation is performed. On the other hand, if in the idle state the Hamiltonian vanishes (a very useful property as outlined in Secs. II.A and II.B), the environment-dominated regime is realized. One has to ensure that the phase coherence rate in this regime is still low enough. B. Spin-boson model Before we proceed to discuss specific physical systems, let us review what is known about the spin-boson model, which has been studied extensively (see reviews by Leggett et al., 1987 and Weiss, 1999). It models the environment as an oscillator bath coupled to one component of the spin. The Hamiltonian reads where HH ctrl z a a x a H B, (4.1) H ctrl 1 B z z 1 B x x (4.) E cos zsin x (4.3) is the controlled part of the Hamiltonian [cf. Eqs. (.3) and (.6)], while p a H B a m a a x a m a (4.4) is the Hamiltonian of the bath. The bath operator X a a x a couples to z. In thermal equilibrium one finds for the Fourier transform of the symmetrized correlation function of this operator X 1 Xt,Xt Jcoth k B T, (4.5) where the bath spectral density is defined by a J a m a a. (4.6) a This spectral density typically has a power-law behavior at low frequencies (Leggett et al., 1987). Of particular interest is Ohmic dissipation, corresponding to a spectrum J, (4.7) which is linear at low frequencies up to some highfrequency cutoff c. The dimensionless parameter reflects the strength of dissipation. Here we concentrate on weak damping, 1, since only this regime is relevant for quantum-state engineering. But still the Hamiltonian-dominated and the environment-

17 Makhlin, Schön, and Shnirman: Quantum-state engineering 373 dominated regimes are both possible depending on the ratio between the energy scale EB z B x, characterizing the coherent evolution, and the dephasing rate (to be determined below). The Hamiltonian-dominated regime is realized when Ek B T. In this regime it is natural to describe the evolution of the system in the eigenbasis (.7) which diagonalizes H ctrl : H 1 E zsin x cos z XH B. (4.8) Two different time scales characterize the evolution (Görlich et al., 1989; Weiss and Wollensak, 1989; Weiss, 1999). On a first, dephasing time scale the offdiagonal (in the preferred eigenbasis) elements of the qubit s reduced density matrix decay to zero. They are represented by the expectation values of the operators (1/)( x i y ). Dephasing leads to the following time dependence (at long times): t 0e iet e t/. (4.9) On the second, relaxation time scale relax the diagonal entries tend to their thermal equilibrium values: z t z z 0 z e t/ relax, (4.10) where z ()tanh(e/k B T). The dephasing and relaxation times were originally evaluated for the spin-boson model in a path-integral technique (Leggett et al., 1987; Weiss, 1999). The rates are 9 1 relax sin E E coth k B T, (4.11) 9 Note that in the literature usually the evolution of z (t) has been studied. To establish the connection to the results (4.11) and (4.1) one has to substitute Eqs. (4.9) and (4.10) into the identity z cos z sin x. Furthermore, we neglect renormalization effects, since they are weak for relax 1 cos k BT. (4.1) In some cases these results can be derived in a simple way, which we present here to illustrate the origin of different terms. As is apparent from the Hamiltonian (4.8) the problem can be mapped on the dynamics of a spin-1/ particle in the external magnetic field E pointing in the z direction and a fluctuating field in the x-z plane. The x component of this fluctuating field, with magnitude proportional to sin, induces transitions between the eigenstates (.7) of the unperturbed system. Applying the golden rule for this term, one obtains readily the relaxation rate (4.11). The longitudinal component of the fluctuating field, proportional to cos, does not induce relaxation processes. It does, however, contribute to dephasing since it leads to random fluctuations of the eigenenergies and thus to a random relative phase between the two eigenstates. As an example we analyze its effect on the dephasing rate in an exactly solvable limit. The unitary operator Uexp i z with a p a (4.13) a m a a transforms the Hamiltonian (4.1) (4.4) to a rotating spin frame (Leggett et al., 1987): H UHU 1 1/E cos z 1/E sin e i H.c.H B. (4.14) Here we recognize that in the limit 0 the spin and the bath are decoupled, which allows an exact treatment. The trivial time evolution in this frame, (t) exp(iet) (0), translates in the laboratory frame to te i(t) e i(0) e iet 0. (4.15) To average over the bath we need the correlator Pte i(t) e i(0) e i(t) e i(0), (4.16) which has been studied extensively by many authors (Leggett et al., 1987; Odintsov, 1988; Panyukov and Zaikin, 1988; Nazarov, 1989; Devoret et al., 1990). It can be expressed as P(t)expK(t), where Kt 4 J d 0 coth k B T cos t1i sin t. (4.17) For the Ohmic bath (4.7) for t/k B T one has Re K(t)(k B T/) t. Thus we reproduce Eq. (4.9) with given by Eq. (4.1) in the limit 0. While it is not so simple to derive the general result for arbitrary, it is clear from Eqs. (4.11) and (4.1) that the effects of the perpendicular (sin ) and longitudinal (cos ) terms in Eq. (4.8) add up independently. In the environment-dominated regime, Ek B T, the qubit s Hamiltonian is too weak to fix the basis, while the coupling to the bath becomes the dominant part of the total Hamiltonian. Therefore one should discuss the problem in the eigenbasis of the observable z to which the bath is coupled. The spin can tunnel incoherently between the two eigenstates of z. To find the tunneling rate one can again use the canonical transformation (4.13) leading to the Hamiltonian (4.14). In the golden rule approximation one obtains (Leggett et al., 1987) the following relaxation rate (for Ek B T): 1 relax E sin 4 P E cos P E cos E sin k B T, (4.18) where P ( ) is the Fourier transform of P(t). To find the dephasing rate we again use Eq. (4.15) and obtain

18 374 Makhlin, Schön, and Shnirman: Quantum-state engineering 1 k B T/. (4.19) In this regime the dephasing is much faster than the relaxation. Moreover, we observe that 1 relax (k B T) 1 while 1 k B T. This implies that the faster is the dephasing, the slower is the relaxation. Such a behavior is an indication of the Zeno (watchdog) effect (Harris and Stodolsky, 198): the environment frequently observes the state of the spin, thus preventing it from tunneling. C. Several fluctuating fields and many qubits Next we consider the general case of a qubit coupled to several fluctuation sources (baths) via different spin components and we discuss below a many-qubit system. It is described by the following generalization of the spin-boson model: HH ctrl i n i a a i x a i i H B i, (4.0) where the index i labels the different (Ohmic) baths and the unit vectors n i determine the spin components to which the baths are coupled. The Hamiltoniandominated regime is realized when E i i k B T, where i correspond to the bath i. In this case one should divide the fluctuations of all baths into transverse and longitudinal ones, as in Eq. (4.8), with each bath being characterized by the angle i between n i and the field direction (cos i Bn i /B). The transverse fluctuations will add up to the relaxation rate as 1 E E relax i sin i coth i k B T, (4.1) while the longitudinal fluctuations lead to the dephasing rate 1 i i cos i k B T 1 1 relax. (4.) In the simplest environment-dominated situation one of the baths (ii 0 ) is much stronger than all others, i0 ii0 i, and satisfies i0 k B TE. Then the dephasing is described in the eigenbasis of n i0 corresponding to this bath. The rate is 1 i0 k B T/. (4.3) The relaxation rate in this basis may be estimated using the golden rule: E sin 1 i0 relax i0 k B T k ii0 i sin B T i, (4.4) where cos i (n i n i0 ). This rate is smaller than the dephasing rate (4.3). Note that the relaxation rates due to baths other than the strongest one do not show the Zeno effect. In the most complicated case of the environmentdominated regime, with several baths coupled to different spin components and characterized by constants i of the same order, it is difficult to make quantitative predictions. However, we expect that the time scale for a quantum-state destruction (either dephasing or relaxation) will be longer than ( i i k B T/) 1. So far we have been concerned with dissipative effects in a single qubit. However, all but the simplest applications of quantum-state engineering make use of many coupled qubits and entangled states, i.e., states whose properties cannot be reduced to the single-bit ones. Therefore the question arises: how does dissipation affect multiqubit systems and entangled states? As a first step we analyze the effect of dissipation on an N-qubit system during an idle period when the single-bit terms and two-bit interactions are switched off, H ctrl 0. We assume that each qubit is coupled to an independent oscillator bath, with no correlations between the baths, and we choose a basis in which this coupling is diagonal, i.e., the bath is coupled to the z component. In this case the time evolution operator governing the density matrix, ˆ 1 N ; 1 N (t), factorizes. We perform for each qubit i the unitary rotation (4.13) with the result 1 N ; 1 N t N i1 e i i (t)( i i ) e i i (0)( i i ). (4.5) Averaging over the baths yields in the long-time limit the following time dependence of the density matrix: 1 N ; 1 N t i:i i expt/ i. (4.6) Here the product is over all qubits i that have offdiagonal entries in the density matrix. This form shows that the dissipation has the strongest effect on those entries of ˆ that are off-diagonal with respect to each qubit. For instance, the dephasing rate of 0 0;1 1 (t) is i 1/ i 1/. It scales linearly with the number of qubits. The result (4.6) applies independent of the initial state, i.e., equally for product states or entangled states of the multiqubit system. However, it is valid only if the controlled parts of the Hamiltonian are switched off. The question of how dissipation influences the dynamics of entangled states in general situations, e.g., during logic operations when the many-qubit Hamiltonian is nonzero, remains open. Further work is needed to analyze this interesting and important problem. D. Dephasing in charge qubits We now turn to the specific case of a Josephson charge qubit coupled to the environment and determine how the dephasing and relaxation rates depend on system parameters. The system is sensitive to various electromagnetic fluctuations in the external circuit and the substrate, as well as to background charge fluctuations. We first estimate the effect of fluctuations originating from the circuit of the voltage sources. In Fig. 13 the

19 Makhlin, Schön, and Shnirman: Quantum-state engineering 375 FIG. 13. A qubit in an electromagnetic environment characterized by the impedance Z(). equivalent circuit of a qubit coupled to an impedance Z() is shown. The latter has intrinsic voltage fluctuations with a Johnson-Nyquist power spectrum. When embedded in the circuit shown in Fig. 13 but with E J 0, the voltage fluctuations between the terminals of Z() are characterized by the spectrum VV ReZ t coth. (4.7) k B T Here Z t ()ic qb Z 1 () 1 is the total impedance between the terminals of Z(), and C qb is the capacitance (.13) of the qubit in the circuit. Following Caldeira and Leggett (1983), we model the dissipative element Z() by a bath of harmonic oscillators described by the Hamiltonian H B as in Eq. (4.4) (Shnirman et al., 1997). The voltage fluctuations between the terminals of Z() are represented by V x, while the spectral function (4.6) has to be chosen as J() ReZ t () in order to reproduce the fluctuation spectrum (4.7). To derive the Hamiltonian we introduce an auxiliary variable, the charge q on the gate capacitor, and add a term that couples q to the bath (a natural choice since the current through the impedance is q ). For simplicity of derivation it is convenient also to add a small inductance in series with the impedance Z to provide a mass for the q mode. Allowing for a time-dependent external voltage V g (t) and integrating out q, we find the Hamiltonian H enc gv g t p E C J C g J cos a a m a a m a a C qb V g t x a a m a a en C qb C J. (4.8) The voltage V g (t) in the last term (coupling to the bath) is relevant only if it depends on time and is usually dropped in the literature. Its role is to provide the RC time delay. This means that when V g is changed the qubit feels the change only after the RC time. In the systems described here this delay is much shorter than the other relevant time scales and thus need not be discussed further. To be specific we shall concentrate in the following on the fluctuations due to an Ohmic resistor Z()R V in the bias voltage circuit. In the two-state approximation, using the relations n(1/)(1 z ) and cos (1/) x, we thus arrive at the Hamiltonian of the spinboson model (4.1) with B z and B x given by Eqs. (.4) and (.5), respectively. The dimensionless parameter z ch, characterizing the effect of fluctuations coupling to z of the charge qubit, is given by z ch 4R V R K C qb C J. (4.9) The circuit resistance is compared to the quantum resistance R K h/e 5.8 k. Since the parameter z ch directly relates the dephasing rate to typical energy scales, its inverse determines the number of coherent singlequbit manipulations that can be performed within the dephasing time. From Eq. (4.9) we see that in order to keep the dissipative effects of external voltage fluctuations weak one has to use a voltage source with low resistance and choose the gate capacitance C g C qb C J as low as possible. The latter screens out the voltage fluctuations, at the expense of having to apply larger gate voltages for the manipulations. At the high frequencies discussed here the typical impedance of the voltage circuit is R V 50, and one obtains z ch 10 (C g /C J ). Taking the ratio 10 C g /C J 10 one can reach a dissipation as weak as z ch 10 6, allowing in principle for 10 6 coherent single-bit manipulations. Fluctuations of the externally controlled flux through the SQUID loop of a charge qubit with tunable Josephson coupling (see Fig. 3) also destroy phase coherence. As can be seen from Eq. (.10), these fluctuations couple to cos x. Analogously to the voltage fluctuations considered above, their strength can be expressed by the effective impedance of the current circuit that supplies the flux. For an estimate we take this impedance R I to be purely real. At typical high frequencies of the qubit s operation it is of the order of the vacuum impedance, R I 100. In terms of this resistance the effect of fluctuations in the flux is characterized by the parameter x ch R K 4R I M E J x 0 x, (4.30) where the flux-controlled Josephson coupling E J ( x )is given by Eq. (.11). The effect is weak for low values of the mutual inductance. For M nh and E J K we obtain x ch The dephasing and relaxation times, if only flux fluctuations need to be con- 10 Nakamura et al. (1999) reached an even smaller ratio for the qubit, but the probe circuit introduced a high stray capacitance.

20 376 Makhlin, Schön, and Shnirman: Quantum-state engineering sidered, are thus given by Eqs. (4.11), (4.1), and (4.30), but with a substitution tan B z /B x since the noise terms couple to x. When both the gate voltage and the flux fluctuate, a multibath situation described by Eq. (4.0) is realized. For typical parameters, e.g., for those of the experiments of Nakamura et al. (1999), one can estimate the dephasing time associated with the noise in the external circuit to be of the order of 100 ns. These experiments allow a direct probing of the phase coherence. Coherent oscillations have been observed for 5 ns. Hence the theoretical estimate appears to be of the right order. 11 Another important source of decoherence in charge qubits is fluctuations of the background charge. It has been found experimentally that they lead to 1/f noise at low frequencies. Typically their contribution to fluctuations of the effective gate voltage, Q g C g V g, is of order S Q ()10 8 e /. These fluctuations limit the time of coherent evolution and can hardly be improved by present-day experimental techniques. 1 Fortunately for the ideas of quantum-state engineering, the configurations of background charges change slowly, over a relaxation time scale that can reach minutes or even hours. Thus during each cycle of manipulations these charges provide random static gate voltages, which do not destroy, for example, coherent oscillations. Their effect can be suppressed using refocusing techniques. When averaging over many experimental runs [as done by Nakamura et al. (1999)], one forms an ensemble average from which the dephasing rate can be extracted. When the qubit is coupled to a measurement device, which necessarily implies a coupling to a macroscopic variable with dissipative dynamics, the feedback introduces fluctuations and causes dephasing. We shall discuss this explicitly in the next section for the case in which a Josephson charge qubit is coupled to a dissipative single-electron transistor. We find that the shot noise of the tunneling current in the SET introduces dephasing and relaxation processes. E. Dephasing in flux qubits Flux qubits have the advantage that they are practically insensitive to background charge fluctuations and 11 In the experiments of Nakamura et al. (1999) much of the dephasing can actually be attributed to the measurement device, a dissipative tunnel junction that was coupled permanently to the qubit. Its tunneling resistance was optimized to be large enough not to destroy the qubit s quantum coherence completely, but low enough to allow for a measurable current. Single-electron tunneling processes, occurring on a time scale of the order of 10 ns, destroy the state of the qubit (escape out of the two-state Hilbert space), thus putting an upper limit on the time when coherent time evolution can be observed. For a more detailed discussion of the experiment and the measurement process we refer to the article by Choi et al. (001). 1 See, however, recent work of Krupenin et al. (000) where the 1/f noise was suppressed by fabricating a metallic island on top of an electrode instead of placing it on the substrate. are advertised for this reason (Mooij et al., 1999). However, their phase coherence can still be destroyed by a number of effects. Some sources of dissipation for flux qubits have been discussed by Tian et al. (000), and estimates have been provided for the parameters of the circuits of Mooij et al. (1999) and Orlando et al. (1999). This includes the effect of background charge fluctuations ( 0.1 s) as well as quasiparticle tunneling in the superconductor with a nonvanishing subgap conductance ( 1 ms). The effect of nuclear spins in the substrate producing fluctuating magnetic fields is similar to the effect of background charges on charge qubits. While the static random magnetic field may induce substantial changes in qubit frequencies of order nucl 30 MHz (which can be suppressed, in principle, by using refocusing pulses), they cause no dephasing until a typical nuclear spin relaxation time T 1, which can reach minutes. Other sources of dephasing studied by Tian et al. (000) include electromagnetic radiation ( 10 3 s), much weaker than in typical rf-squid designs, and unwanted dipole-dipole magnetic couplings between qubits, which for an interqubit distance of 10 m produces substantial effects after a relatively short time 0. ms. Design variations have been suggested to reduce the latter effect. An important source of flux qubit dissipation is fluctuation in the external circuit that supplies fluxes through the loops. This can be analyzed along the same lines as presented above. Since such fluctuations couple to z, the relevant parameter is z fl. It is fixed by the impedance Z I of the current source in the input loop providing the flux bias and by the mutual inductance of the input and the qubit s loop M:. (4.31) fl z a 4 R K Re Z I 1 4 E J M 0 The numerical prefactor in Eq. (4.31) is a6( L 1)/ for an rf SQUID and ab 4b 1/ for the design of Mooij et al. (1999), with bẽ J /E J being the ratio of critical currents for the junctions in the loop (see Sec. III.A). The dephasing is slow for small loops and junctions with low critical currents. Indeed, the argument in brackets in Eq. (4.31) can be represented as the product of the screening ratio M/L and the quantity L 4 LE J / 0. While the latter should be larger than one for an rf SQUID, it can be much smaller for the design of Mooij et al. (1999), resulting in a slower dephasing. The impedance Z I () and the bath spectrum J() Re Z I 1 () are frequency dependent. They can lead to resonances, for instance, if the current source is attached to the qubit via low-loss lines. Care has to be taken in experiments to avoid these resonances. In addition, the dephasing and relaxation times have to be estimated for this situation. The analysis of the Hamiltonian (4.8) with a general bath spectrum shows that the dephasing time can be defined as the time for the quantity cos ( 0 t X()d) to reach a value of order one, while the relaxation time is the time for

21 Makhlin, Schön, and Shnirman: Quantum-state engineering 377 sin t 0 X()e ie d to reach this value. The first quantity can be expressed as tdx () involving the function () sin (t/)/(t ), which is peaked at 0 and has width t 1. Recalling that X is related to the bath spectrum (4.5), one finds that only the low-frequency part of the impedance Z I (), with 1, determines the dephasing rate. The relaxation rate depends on the values of Z I () at frequencies in the vicinity of E/ over a range of width 1 relax. The choice of a high dc resistance for the remote current source strongly suppresses fluctuations at low frequencies. Using Eqs. (4.31) and (4.1) we can estimate the dephasing rate for the parameters of Mooij et al. (1999). Assuming Z I (0)1 M and ML we find fl z 10 9, which implies a negligible dephasing. At higher frequencies of order E, even if resonances are suppressed, Z I (E) is of order 100, leading to fl z This determines the relaxation rate via Eq. (4.11). For Mooij et al. s (1999) parameters we estimate a relaxation time of 5 s at the degeneracy point. The effect of fluctuations of the x term in the Hamiltonian (in the flux circuit of the dc-squid loop that controls the Josephson coupling) can be described in a similar way (Makhlin et al., 1999). The effect of these fluctuations is relatively weak for the operation regimes discussed by Mooij et al. (1999). While a further analysis of the dephasing effects in flux qubits may be needed, the above-mentioned estimates of suggest that the observation of coherent flux oscillations is feasible in the near future. However, the problem remains that observation requires an efficient quantum detector, e.g., a quantum magnetometer. We discuss this issue in the next section. V. THE QUANTUM MEASUREMENT PROCESS A. General concept of quantum measurements Quantum-state engineering requires not only controlled quantum manipulations but also quantum measurement processes. They are needed, for example, at the end of a computation to read out the final results, or even in the course of the computation for the purpose of error correction. The problem of quantum measurement has always attracted considerable attention, and it still stirs up controversy. In most of the literature on quantum information theory the measurement process is expressed simply as a wave-function collapse, i.e., as a nonunitary projection, which reduces the quantum state of a qubit to one of the possible eigenstates of the observed quantity with state-dependent probabilities. On the other hand, in reality any measurement is performed by a device that itself is a physical system, suitably coupled to the measured quantum system and with a macroscopic readout variable. This is accounted for in the approaches described below, which are based on the concepts of dissipative quantum mechanics (Leggett et al., 1987; Zurek, 1991; Weiss, 1999). The qubit and the measuring device are described as coupled quantum systems. Initially there exist no correlations between the qubit and the meter but, due to the coupling, such correlations (entanglement) emerge in time. This is precisely the concept of measurement of von Neumann (1995). Furthermore, the meter is actually a dissipative system, coupled to the environment. The dissipation eventually reduces the entanglement between the qubit and the meter to classical correlations between them. This is exactly what is needed for the measurement. Several groups have studied problems related to quantum measurement processes in mesoscopic systems. Aleiner et al. (1997), Gurvitz (1997), and Levinson (1997), investigated the effect of a dissipative conductor, whose conductance depends on the state of the quantum system, on this system itself. Such a configuration was realized in recent experiments (Buks et al., 1998; Sprinzak et al., 000), in which a quantum dot was embedded in one of the paths of a which-path interferometer. The flow of a dissipative current through a nearby quantum point contact, with conductance that depends on the charge on the dot, amounts to a measurement of the path chosen by the electrons in the interferometer. And indeed the flow of a current through the point contact leads to an observable reduction in the flux dependence of the current through the interferometer. On the other hand, no real measurement was performed. The experiments merely demonstrated that the amount of dephasing could be controlled by a dissipative current through the quantum point contact, but they did not provide information about the path chosen by each individual electron. In what follows we shall discuss systems in which one is able not only to study the dephasing but also, for instance, by measuring a dissipative current, to extract information about the quantum state of the qubit. As an explicit example we investigate a single-electron transistor (SET) in the sequential tunneling regime coupled capacitively to a charge qubit (Schoelkopf et al., 1998; Shnirman and Schön, 1998). Our purpose is not philosophical but rather practical. We do not search for an ideal measurement device; instead we describe the properties of a realistic system, known to work in the classical regime, and investigate whether it can serve as a quantum measurement device. At the same time we keep in mind that we should come as close as possible to the projective measurement picture assumed in quantum algorithms. Various other mesoscopic devices have been studied recently with the same goal, namely, to establish their potential use as quantum detectors. These include a quantum point contact coupled to a quantum dot (Gurvitz 1997; Korotkov, 1999), a single-electron transistor in the cotunneling regime (Averin, 000a; Maassen van den Brink, 000), or a superconducting SET (SSET; Averin, 000b; Cottet et al., 000). For a flux qubit a dc SQUID, or suitable modifications of it, can serve as a quantum detector. Technically the measurement process is described by the time evolution of the reduced density matrix of the coupled system of qubit and meter. To analyze it we first

22 378 Makhlin, Schön, and Shnirman: Quantum-state engineering FIG. 14. The circuit consisting of a qubit plus a single-electron transistor used as a measuring device. derive a Bloch master equation for the time evolution of the system of a Josephson charge qubit coupled to a dissipative SET. This derivation demonstrates how the unitary time evolution of system plus detector can lead to a quantum measurement. It also allows us to follow the dynamics of the system and detector and to analyze the mutual influence of their variables. During the measurement the qubit loses its phase coherence over a short dephasing time. 13 This means that the offdiagonal elements of the qubit s density matrix (in a preferred basis, which depends on relative strengths of the coupling constants) vanish, while the diagonal elements remain unchanged. At the same time the information about the initial state of the qubit is transferred to the macroscopic state of the detector (the current in the SET). Under suitable conditions, after another time meas, this information can be read out. Finally, over a longer, mixing (or relaxation) time scale mix, the detector acts back onto the qubit and destroys the information about the initial state. The diagonal entries of the density matrix tend to their stationary values. (These are either determined by the detector or thermally distributed, depending on the relative strength of the measurement device and residual interactions with the bath.) One has to choose parameters such that this back action does not change the occupation probabilities of the qubit s state before the information is actually read out, meas. 13 In this section denotes the dephasing time during a measurement. It is usually much shorter than the dephasing time during the controlled manipulations discussed in the previous sections. From the context it should be clear which situation we refer to. The different times scales characterizing the measurement process by a SET also show up in stationary state properties, e.g., in the noise spectrum of the fluctuations in the SET. This can be extracted from the time evolution of the coupled density matrix as well. Ideally, the meter is coupled to the qubit only during the measurement. In practice, however, this option is hard to realize for mesoscopic devices. Instead, the meter and the qubit are coupled permanently but the former is kept in a nondissipative state. To perform a measurement, the meter is switched to the dissipative regime, which, as we mentioned above, is an important requirement for effective quantum measurement. For instance, in a SET, due to the Coulomb blockade phenomena, no dissipative currents are flowing in the meter as long as the transport voltage is switched off. Applying a voltage bias above the Coulomb threshold induces a dissipative current through the SET, which leads to dephasing and, at the same time, provides the macroscopic readout variable. Various modes of operation can be used, depending on details of a particular setting. For instance, one can switch the meter abruptly into the dissipative regime and monitor the response of dissipative currents to the qubit. Another possibility is to change the bias gradually until the system switches into the dissipative regime. The value of the bias at which switching occurs provides information about the qubit s state. Conceptually these techniques are similar. For definiteness, below we discuss the former operation strategy. B. Single-electron transistor as a quantum electrometer Since the relevant quantum degree of freedom of a Josephson charge qubit is the charge of its island, the

23 Makhlin, Schön, and Shnirman: Quantum-state engineering 379 natural choice of measurement device is a singleelectron transistor. This system is shown in Fig. 14. The left-hand part is the qubit, with state characterized by the number of extra Cooper pairs n on the island and controlled by its gate voltage V qb g. The right-hand part shows a normal island between two normal leads, which form the SET. Its charging state is characterized by the number of extra single-electron charges N on the middle SET island, controlled by gate and transport voltages, V g and V tr, and, due to capacitive coupling to the qubit, further controlled by the state of the latter. A similar setup has been studied in the experiments of Bouchiat (1997) and Bouchiat et al. (1998), in which it was used to demonstrate that the ground state of a single-cooperpair box is a coherent superposition of different charge states. During the quantum manipulations of the qubit the transport voltage V tr across the SET transistor is kept at zero and the gate voltage of the SET, V SET g, is chosen to tune the island away from degeneracy points. Therefore at low temperatures Coulomb blockade effects suppress exponentially a dissipative current flow in the system, and the transistor merely modifies the capacitances of the system. 14 To perform a measurement one tunes the SET by V SET g to the vicinity of its degeneracy point and applies a small transport voltage V tr. The resulting normal current through the transistor depends on the charge configuration of the qubit, since different charge states induce different voltages on the island of the SET. While these properties are well established as the operation principle of a SET as electrometer in the classical limit, it remains to be demonstrated that they also allow the resolution of different quantum states of the qubit. For this purpose we have to discuss various noise factors, including the shot noise associated with the tunneling current and the measurement-induced transitions between the states of the qubit. These can be accounted for by analyzing the time evolution of the combined system s density matrix. The Hamiltonian of the combined system consists of the parts describing the qubit, the SET, and the interaction between them: HH ctrl H SET H int. (5.1) Except for a redefinition and renormalization of parameters, the qubit s part is identical to that of the bare qubit (.3), H ctrl 1 B zˆ z 1 B xˆ x 14 More precisely, the leading contributions are cotunneling processes, which are weak in high-resistance junctions. with B z 4E C,qb (1n g,qb ) and B x E J ( x ). For a detailed derivation and precise definition of the parameters see Appendix C. Here it is sufficient to know that the qubit s Hamiltonian can be controlled by gate voltages and the flux through the SQUID. After diagonalization of Eq. (.8), H ctrl (1/)E() z, the characteristic energy scale of the qubit, E()B z B x, becomes apparent. The Hamiltonian of the SET reads H SET E C,SET NN g,set H L H R H I H T, (5.) where the transistor s charging energy is given by Eq. (C) and the gate charge N g,set ev N /E C,SET, defined in Eq. (C4), can be controlled by V SET g. The three terms H L, H R, and H I describe microscopic degrees of freedom of noninteracting electrons in the two leads and the middle island of the SET: H r k r k c r r k c k rl,r,i. (5.3) The index rl,r,i labels the electrodes left, right (viewed from a suitable angle), and island, labels transverse channels including the spin, while k refers to the wave vector within one channel. Note that similar terms should have been written for the electrode and island of the qubit; however, for this superconducting nondissipative element the microscopic degrees of freedom can be integrated out (Ambegaokar et al., 198; Schön and Zaikin, 1990), resulting in the macroscopic quantum description presented in Secs. II and III. In this limit the tunneling terms reduce to the Josephson coupling H J E J cos, expressed in a collective variable describing the coherent transfer of Cooper pairs, e i nn1. The normal-electron tunneling in the SET is described by the standard tunneling Hamiltonian, which couples the microscopic degrees of freedom: H T kk kk L T kk c k I L c k e i R T k kc R I k c k e i e i H.c. (5.4) To make the charge transfer explicit, Eq. (5.4) displays two macroscopic operators, e i and e i. The first describes changes in the charge on the transistor island due to tunneling: e i NN1. If the total number of electrons on the island is large it may be treated as an independent degree of freedom. We further include the operator e i, which acts on m, the number of electrons that have tunneled through the SET, e i mm1. Since the chemical potential of the right lead is controlled, m does not appear in any charging part of the Hamiltonian. However, we have to keep track of it, since it is the measured quantity, related to the current through the SET. Finally, H int E int N(n1) describes the capacitive interaction between the charge on the qubit s island and that on the SET island. In detail it originates from the mixed term in Eq. (C1), where E int is given by Eq. (C). In the two-state approximation for the qubits, n(1 z )/, we obtain H int NH int NE int z. (5.5)

24 380 Makhlin, Schön, and Shnirman: Quantum-state engineering FIG. 15. P(m,t), the probability that m electrons have tunneled during time t (measured in nanoseconds). The initial amplitudes of the qubit s states are a 0.5, b E J 0.1 K; the remaining parameters are given later in the text. The operator H int, introduced here for later convenience, is the part of the interaction Hamiltonian that acts in the qubit s Hilbert space. C. Density matrix and description of measurement The total system composed of qubit and SET is described by a total density matrix ϱˆ (t). We can reduce it, by tracing out the microscopic electron states of the left and right leads and of the island, to ˆ ttr L,R,I ϱˆ t. (5.6) This reduced density matrix ˆ (i,i;n,n;m,m) is still a matrix in the indices i, which label the quantum states of the qubit 0 or 1, inn, and in m. In the following we shall assume that initially, as a result of previous quantum manipulations, the qubit is prepared in some quantum state and is disentangled from the SET, i.e., the initial density matrix of the whole system may be written as a product ˆ 0ˆ 0qb ˆ 0SET. At time t0 we switch on a transport voltage of the SET and follow the resulting time evolution of the density matrix of the whole system. For specific questions we may further reduce the total density matrix in two ways, either of which can provide complementary information about the measurement process. The first, common procedure (e.g., Gurvitz, 1997) is to trace over N and m. This yields a reduced density matrix of the qubit ˆ ij N,m ˆ (i,j;n,n;m,m). At t0 one has ˆ ij (ˆ 0qb ) ij. Depending on the relation among the energy scales and coupling strengths (see Sec. IV), a preferred basis may exist in which the dynamics of the diagonal and the off-diagonal elements of ˆ ij decouple. We study how fast the off-diagonal elements (in that special basis) vanish after the SET is switched to the dissipative state, i.e., we determine the rate of dephasing induced by the SET. And we determine how fast the diagonal elements change their values. This process we call mixing. The rates of dephasing and mixing refer to the quantum properties of the measured system in the presence of the measurement device. These quantities have also been analyzed by Aleiner et al. (1997), Gurvitz (1997), Levinson (1997), and Buks et al. (1998). They do not tell us anything, however, about the quantity measured in an experiment, namely, the current flowing through the SET. Therefore the second way to reduce the density matrix is important as well. By tracing the density matrix over the qubit s variables and the state N of the island, Pm,t ˆ i,i;n,n;m,mt, (5.7) i,n we obtain a probability distribution for the experimentally accessible number of electrons m that have tunneled through the SET during time t. A detailed analysis will be presented below. In order to provide a feeling for this procedure we first present and describe some representative results: At t0 no electrons have tunneled, so P(m,0) m,0. Then, as illustrated in Fig. 15, the peak of the distribution moves to nonzero values of m and, simultaneously, widens due to shot noise. If the two states of the preferred basis correspond to different tunneling currents, and hence different m shifts, and if the mixing in this basis is sufficiently slow, then after some time the peak splits into two with weights corresponding to the initial values of the diagonal elements of ˆ 0qb in the preferred basis (we shall denote these values a and b 1a ). Provided that after sufficient separation of the two peaks their weights are still close to the original values, a good quantum measurement can be performed by measuring m. After a longer time, due to transitions between the states of the preferred basis (mixing), the two peaks transform into a broad plateau. Therefore there is an optimum time for the measurement such that, on the one hand, the two peaks are separate and, on the other hand, the mixing has not yet influenced the process. Having introduced the relevant time scales, we can mention that which-path interferometry (Buks et al., 1998) can be thought of as a very short measurement process. Each particular electron the observed quantum system spends only a short time within the dot. This time of interaction with the meter may be shorter than the dephasing time. Therefore the coupling leads only to a slight suppression of the interference pattern.

25 Makhlin, Schön, and Shnirman: Quantum-state engineering 381 FIG. 16. The Dyson-type equation governing the time evolution of the density matrix. It is equivalent to the generalized master eqation (5.8). The self-energy diagrams describe transitions due to tunneling in the SET transistor. D. Master equation The time evolution of the density matrix leads to Bloch-type master equations with coherent terms. Examples of this type have recently been analyzed in various contexts (Nazarov, 1993; Schoeller and Schön, 1994; Gurvitz and Prager, 1996; Stoof and Nazarov, 1996; Gurvitz, 1997). Schoeller and Schön (1994) developed a diagrammatic technique that provides a formally exact master equation for a SET as an expansion in the tunneling term H T, while all other terms, including the charging energy, constitute the zeroth-order Hamiltonian H 0. The time evolution of the reduced density matrix, ˆ (t) ˆ (0)(0,t), is expressed by a propagator (t,t), which is expanded and displayed diagrammatically and finally summed in a way reminiscent of a Dyson equation. Examples are shown in Fig. 16 and Appendix D. In contrast to equilibrium many-body expansions, since the time dependence of the density matrix is described by a forward and a backward time-evolution operator, there are two propagators, which are represented by two horizontal lines (Keldysh contour). The two bare lines describe the coherent time evolution of the system. They are coupled due to the tunneling in the SET. The sum of all distinct transitions defines a self-energy diagram. Below we shall present the rules for calculating and present a suitable approximate form. The Dyson equation is equivalent to a Bloch master equation for the density matrix, which reads dˆ t i dt ˆ t 0 t,h dtˆ ttt. (5.8) 0 In principle, the density matrix ˆ (i,i;n,n;m,m) ˆ i,n,m i, N, m is a matrix in all three indices i, N, and m, and the (generalized) transition rates due to singleelectron tunneling processes (in general of arbitrary order), i,n,m ī, N, m (tt), connect these diagonal i,n,m ī,n,m and off-diagonal states. But a closed set of equations for the time evolution of the system can be derived (Schoeller and Schön, 1994) which involves only the elements of ˆ diagonal in N. The same is true for the matrix structure in m. That is, we need only consider the elements of the density matrix ˆ Nij (m)ˆ i,n,m j,n,m. Accordingly, of all the transition rates we need only calculate the corresponding matrix elements i,n,m i,n,m j,n,m j,n,m(t). If the temperature is low and the applied transport voltage not too high, the leading tunneling processes in the SET are sequential transitions between two adjacent charge states, say, N0 and N1. We concentrate here on this case (to avoid confusion with the states of the qubit we continue using the notation N and N1). The transition rates can be calculated diagrammatically in the framework of the real-time Keldysh contour technique. The derivation is presented in Appendix D. A simplification emerges, since the coherent part of the time evolution of the density matrix, which evolves over the time scale set by the qubit energies, is slow compared to the tunneling time, given by the inverse of the energy transfer in the tunneling process [see arguments in square brackets in Eqs. (5.11) and (5.1) below]. As a result the self-energy effectively reduces to a delta function, (t)(t), and the Bloch type master equation (5.8) reduces to a Markovian dynamics. The resulting master equation is translationally invariant in m space. Hence a Fourier transformation is appropriate, ˆ Nij (k) m e ikm ˆ Nij (m). As a result Eq. (D8) factorizes in k space and we get a finite rank (88) system of equations for each value of k. This system may be presented in compact form if we combine the eight components of the density matrix into a pair (ˆ N,ˆ N1 )of matrices ˆ Nij (k), corresponding to N and N1: dt d ˆ N ih ˆ N1 ctrl,ˆ N ih ctrl H int,ˆ N1 ˇ L ˇ L e ik ˇ R ˇ R ˆ N ˆ N1. (5.9) x The operator H int E int z E int z E int was introduced in Eq. (5.5). The tunneling rates in the left and right junctions are represented by operators ˇ L and ˇ R, acting on the qubit s density matrix: ˇ Lˆ N L ˆ N L H int,ˆ N, ˇ Rˆ N1 R ˆ N1 R H int,ˆ N1. (5.10) Here L/R R K /(4 R T L/R ) is the tunneling conductance of the left/right junction, measured in units of the resistance quantum R K h/e. The tunneling rates in the junctions are determined by the potentials L/R of the leads and the induced charge N g,set on the SET s island: L L L 1N g,set E C,SET, (5.11) R R 1N g,set E C,SET R, (5.1) L R / L R. (5.13) These combine into the parameter, which gives the conductance of the SET in the classical (e.g., highvoltage) regime. As a result of the last terms in Eqs. (5.10) the effective rates and thus the current in the SET are sensitive to the state of the qubit, which makes the measurement possible. To provide a feeling for the types of solutions, we present here numerical solutions of Eq. (5.9) for the following system parameters: B z K, E int 0.5 K, L R 0.03, L 1.8 K, and R 7.8 K. We plot the results for E J 0.1 K (Fig. 15) and for E J 0.5 K (Fig. 17).

26 38 Makhlin, Schön, and Shnirman: Quantum-state engineering FIG. 17. P(m,t), the probability that m electrons have tunneled during time t. The parameters are the same as in Fig. 15 except that E J 0.5 K. We see that for the smaller value of E J (the eigenbasis of H ctrl is closer to the charge basis) the probability distribution P(m,t) develops a two-peak structure. The weights of the peaks are equal to the initial values of the diagonal elements of the qubit s density matrix. For the larger value of E J the two-peak structure can also be seen but the valley between the peaks is filled. This indicates that mixing transitions take place on the same time scale as that of peak separation, and no good measurement can be performed. In Fig. 18 the probability distribution P(m,t) is plotted for longer times. (In order to cover many decades of the time it is necessary to rescale the m axis as well.) The parameters are the same as in Fig. 15 (E J 0.1 K). The figure displays clearly the measurement stage and the third stage on the mixing time scale. The valley between the two peaks fills up and a single, broad (note the rescaled m axis) peak develops. As one can see from Eq. (5.9) the Hamiltonian of the qubit switches between H ctrl and H ctrl H int after each tunneling event in the SET. This leads, in general, to a complicated dynamics. Above we showed representative numerical results. Next we shall analyze Eq. (5.9) perturbatively, which provides insight into the physics of the measurement process. In general, the two Hamiltonians H ctrl and H ctrl H int are not close and their respective eigenbases may be quite different. In this situation a better choice for the qubit s Hamiltonian would be the average Hamiltonian H av H ctrl NH int, (5.14) where N L /( L R ) is the steady-state average of N. (The left-right asymmetry follows from the specific choice of sequential tunneling regime: N0 N1 in the left junction and N1 N0 in the right junction.) Relevant energy scales are, then, the level splitting of the average Hamiltonian, H av 1 E av z, (5.15) the capacitive coupling energy E int, and the bare tunneling rates L/R. For definiteness we consider the case E int L R. (5.16) That is, we exclude the regimes of too strong coupling or too weak tunneling in the SET, which could be treated similarly to what is presented below. The main simplification of the regime (5.16) is that the fluctuation spectrum of the electron number N on the SET island is white over a rather wide range of frequencies, characterized by the zero-frequency noise power S N ( 0) N 0. When the SET is detached from the qubit and switched to the dissipative regime, this spectrum is given by S N ( 0)4/( L R ). The related backaction noise randomizes the relative phase between the charge states of the qubit. In the absence of other sources for dynamics the rate of this process is 1 0 E int S N 0 4E int L R. (5.17) This is an important measure for distinguishing between different subregimes of Eq. (5.16), which will be described in the following. E. Hamiltonian-dominated regime FIG. 18. P(m,t) obtained in the two-mode approximation (5.33), plotted vs t (on a logarithmic scale) and m/t. The initial probabilities of the qubit s states are a 0.75, b 0.5. Let us first consider the case in which the qubit s average Hamiltonian dominates over the dephasing due to the back action of the SET, E av 1 0. Then a perturbative analysis in the eigenbasis of H av is appropriate, where the operator H int can be rewritten as H int E int z E int x, where E int E int cos av, E int E int sin av, and sin av E J /E av. In the following

27 Makhlin, Schön, and Shnirman: Quantum-state engineering 383 we treat perturbatively the off-diagonal part of H int, i.e., E int x (which leads to mixing; see below). In zeroth order, i.e., for E int 0, the time evolutions of ˆ Nij ij (k),ˆ N1 (k) with different (i,j) are decoupled from each other. For diagonal elements, ij0 or1,we obtain ˆ N1 00/11 d dt ˆ N 0/1 L 0/1 L where 0/1 L L L E int e ik R 0/1 R 0/1 ˆ N ˆ N1 00/11, (5.18) and 0/1 R R R E int. We obtain four eigenmodes (e t ), two for each element (00 or 11). Most interesting are small values k 1. In this case, two modes with eigenvalue ii (k) i i k 1 f i i k and eigenvectors V ii (k) given by ˆ Nii ii (k)/ˆ N1 (k) i i R / L describe waves in m space propagating with group velocity i L i R i L i R i. (5.19) The wave packets widen with time as f i i t due to shot-noise effects. The so-called Fano factors, f i i i L R / i L i R, (5.0) determine the suppression of shot noise in the sequential tunneling regime. The second pair of eigenmodes decays quickly, ii (k)( i L i R ). For their eigenvectors we obtain ˆ Nii ii (k)/ˆ N1 (k)1. This fast decay means that after a few tunneling events, over a time ( i L i R ) 1, detailed balance ˆ Nii ii (k)/ˆ N1 (k) i i R / L is established. For larger values of k (k1) one can check that both ˆ Nii ii (k) and ˆ N1 (k) decay quickly. For the off-diagonal elements we obtain d dt ˆ N ˆ N1 01/10 L e ik R L R ie N 0 0 ie N1 ˆ N ˆ N1 01/10, (5.1) where E N E av NE int, and E N1 E av (1N)E int. Again, there are two pairs of eigenmodes with eigenvalues such that 10 (k) 01 (k)*. One pair of eigenmodes with Re 01/10 (k1)( L R ) decays quickly. For the second pair, 10/01 (k1) ie av 1 where ˆ N1 1 4E int L R E int S N cos av. (5.) Thus, after detailed balance is established, the 10 and 01 modes describe coherent oscillations of the offdiagonal matrix elements with frequency of order E av and decay rate 1. For larger values of k the decay times are of the same order or shorter than those for k 0. The fast-decaying diagonal modes 00/11 do not contribute to P(m,t), since for these modes ˆ Nii (k) ii (k)0. Thus there are only two modes contributing, with eigenvalues 00 (k) and 11 (k). Starting from the initial occupation probabilities a and b we obtain Pk,ta e 00 (k) t b e 11 (k) t, (5.3) where P(m,t)dk/() P(k,t) e ikm. This form describes the evolution of the distribution P(m,t) from the initial (m) att0 into two peaks. The peaks shift to positive m values linear in t with velocities 0 and 1, and they grow in widths as f i i t. This time dependence implies that only after a certain time, which we denote as the measurement time meas, the two peaks emerge from the broadened distribution. The associated rate is 1 meas 0 1 f 0 0 f 1 1. (5.4) In the linear-response regime, when 0 and 1 are close, we obtain 1 meas I, (5.5) 4S I where S I is the zero-frequency power of the shot noise in the SET and Ie( 0 1 ). The weights of the peaks are given by the initial weights of the eigenstates of H av, a and b. Measuring the charge m thus constitutes a perfect quantum measurement (Shnirman and Schön, 1998). Next we analyze the effect of the perturbation E int z in the master equation (5.9). It appears in both the coherent left-hand side and incoherent right-hand side parts of Eq. (5.9) and leads to mixing. As usual the perturbation has the strongest effect when it lifts a (near) degeneracy, since in this case the eigenvectors within the degenerate subspace may change substantially. Therefore we first treat the two near-degenerate modes, i.e., we restrict ourselves to the two-dimensional subspace spanned by V 00 (k0) and V 11 (k0), and check how the degeneracy between these two modes is lifted in a second-order perturbative expansion. To account for this we approximate the diagonal part of the density matrix as ˆ diag k,ta 0 k,tv 00 k0a 1 k,tv 11 k0 (5.6) (the left-hand side should be understood as an eightcolumn vector consisting of all matrix elements of ˆ diag ), for which we obtain an effective (reduced) master equation: d dt A0 A k 1 M red A0 A k 1, (5.7) where M red 00 k k 1 mix 1 1. (5.8) 1 1 The second term in Eq. (5.8) results from the perturbative expansion and, indeed, lifts the degeneracy between

28 384 Makhlin, Schön, and Shnirman: Quantum-state engineering the two modes. The mixing rate is obtained in a secondorder perturbation expansion as 1 mix 4E int E av R L E int S N E av. (5.9) In approximation (5.6) we have A i (k,t) N ˆ Nii (k). At t0 these are occupation probabilities of the eigenstates of H av, A 0 (k,t0)a, A 1 (k,t0)b, for all k. It is straightforward to diagonalize Eq. (5.8) to find the eigenvalues. While the sum of the occupation probabilities is conserved, A 0 (k0,t)a 1 (k0,t)1, the difference decays, A 0 (k0,t)a 1 (k0,t)exp(t/ mix ). Thus both occupations tend to 1/ in the long-time limit. This implies that the evolution of P(m,t) is given by Eq. (5.3) only for times t mix, and in order to perform a 1 measurement one must have mix meas. As meas E int cos av and 1 mix E int sin av, a good measurement may always be achieved by choosing tan av small enough. Another effect of the mixing perturbation is that the dephasing rate, i.e., the decay rate of the off-diagonal elements of the density matrix, is changed as cos av 1/ mix. (5.30) This is analogous to the relation between the dephasing (4.1) and relaxation (4.11) rates in the spin-boson model. It reflects the fact that the relaxation of the diagonal elements of the density matrix leads also to additional suppression of the off-diagonal elements. Thus even when av /, the off-diagonal elements vanish with the rate 1/ mix. The long-time behavior of the qubit and detector, excluding the period of the initial dephasing, is dominated by the two slowly decaying modes. In this regime we obtain from Eqs. (5.7) and (5.8) the reduced time evolution operator U red k,texpm red k t. (5.31) Its Fourier transform U red (m,t) yields the expression for the distribution. (5.3) Pm,t1, 1 U red m,t a b This Fourier transform can be performed analytically (Makhlin et al., 000d), and we arrive at where and Pm,t m P m,tppl m t, t e m /f t P mm,t, (5.33) f t t mix e t/ mixa m 0 tb m 1 t (5.34) 0/1 /. (5.35) We observe that the solution is constructed from two delta peaks, smeared by the convolution with the shotnoise Gaussian, and a plateau between them, which for x1 is given by 1 P pl x,e mix I 0 1x 1xa b I 1 1x / 1x, (5.36) while P pl 0 for x1. Here I 0, I 1 are modified Bessel functions. We work in the limit of weak qubit-detector coupling, where the Fano factors (5.0) are close, and we denote them simply by f. For short times, t mix, the peaks dominate, while the plateau is low. At longer times the initial peaks disappear while the plateau is transformed into a single central peak (see Fig. 18). We complement the results for the charge distribution function P(m,t) by a derivation of the distribution function for values of the current. The measured quantity is actually the current averaged over a certain time interval t, i.e., I t m(tt)m(t)/tm/t. Accordingly the quantity of interest P(I t,t) can be expressed by the joint probability that m electrons have tunneled at time t and mm electrons at a later time tt: PI t,t m Pm,t;mm,tt m Tr Um,t Um,tˆ 0. (5.37) Here we make use of the Markovian approximation (see Sec. V.D). The trace is taken over all degrees of freedom except m, and ˆ 0 refers to their initial state. In the two-mode approximation (5.7) and (5.8), sufficient at long times, we again replace U by U red and obtain PI t,t1 1 U red m,t a b e mix t/ 1 (5.38) 1. The behavior of P(I t,t) is displayed in Fig. 19 for various values of t: (a) If the current is averaged over very short intervals, t meas [Fig. 19(a)], the detector does not have enough time to extract the signal from the shot-noise-governed background. (b) An effective quantum measurement is achieved if meas t mix. In this case the qubit-sensitive signal can be seen on top of the shot noise. Hence, as seen in Fig. 19(b), the measured value of I t is close to either e 0 or e 1 [Eq. (5.19)]. The corresponding probabilities (the weights of the two peaks) are initially a and b. They change at longer times, t mix, to a 1/-1/ distribution due to mixing. A typical current pattern is a telegraph signal jumping between e 0 and e 1 over a time mix. (c) and (d) If the

29 Makhlin, Schön, and Shnirman: Quantum-state engineering 385 FIG. 19. Probability distribution P(I t,t) of the current averaged over various time intervals t (a) (d). The time axis is plotted on a logarithmic scale. current is averaged over longer times t mix, the meter-induced mixing erases the information about the qubit s state. Several telegraph jumps can occur over this time scale, and for t mix one measures only the timeaveraged current between e 0 and e 1, completely insensitive to the initial values. This is shown in Fig. 19(d), while Fig. 19(c) displays the crossover between (b) and (d). Notice that, since P(I t,t0)p(mit,tt), the zero-time limits of the surfaces in Figs. 19(a) (d) are given by the charge distribution function plotted in Fig. 18. We point out that in the Hamiltonian-dominated regime the current in the SET is sensitive to the occupation probabilities of the eigenstates of H av, rather than those in the basis of charge states. This may appear surprising, since the SET couples to the charge operator of the qubit. More precisely, the current is only sensitive to the expectation value of the charge in each eigenstate. As a consequence, at the degeneracy point av /, where the two eigenstates have the same average charge, both eigenstates lead to the same current in the SET and no measurement is possible. The measurement is effective only when charges in the eigenstates differ, tan av 1. F. Detector-dominated regime When back-action dephasing dominates over the average Hamiltonian, 1 0 E av, a perturbative analysis in the charge basis is appropriate. In this basis the perturbation is the Josephson term of the qubit s Hamiltonian H ctrl, i.e., (1/)B x x. Starting from zeroth order, B x E J 0, we obtain equations similar to Eqs. (5.18) and (5.1) with the replacements E av B z,av and E int E int, where B z,av B z NE int is the charging energy of the average Hamiltonian. The analysis of the diagonal and off-diagonal modes is performed similarly. We get for the dephasing rate For the measurement time we reproduce Eq. (5.4). The dynamics of the two long-lived diagonal modes can again be reduced to Eqs. (5.3), (5.6), (5.7), and (5.8) with 1 mix E J 0. (5.39) This result is standard for the Zeno regime, i.e., the regime in which coherent oscillations are overdamped by dephasing [cf. Eq. (4.18)]. The condition for the Zeno regime given above requires a rather strong dephasing, such that 1 0 exceeds 1 both components of the qubit s Hamiltonian (.3), 0 E J,B z,av. The second part of this condition can be satisfied by tuning the qubit close to the degeneracy point. In contrast, in the Hamiltonian-dominated regime it is desirable for a good measurement to switch the qubit away from the degeneracy point. The long-time behavior of the charge and current distributions, P(m,t) and P(I t,t), is again given by Eqs. (5.33) and (5.38) but with proper redefinitions of 0, 1, 1 mix, etc., which now refer to the detector-dominated regime. In particular, the measurement provides information about the initial occupations of the qubit charge states rather than the eigenstates. To summarize we present the main results for the two regimes in Table I.